{
  "arm": "Opus 4.8",
  "provider": "anthropic",
  "benchmark": "postcutoff_pilot_v1",
  "freeze_tag": "postcutoff-pilot-v1-freeze-20260705",
  "note": "Frontier comparator. Single public model; full per-task output included.",
  "tasks": [
    {
      "task_id": "PILOT-N2",
      "task_type": "math",
      "answer": "30907",
      "gold": "30907",
      "correct": true,
      "tokens_in": 219,
      "tokens_out": 17291,
      "cost_usd": 0.43337,
      "latency_s": 162.93,
      "timed_out": false,
      "solution": "<think>\nWe need to tile a 3\u00d710 rectangle using exactly two 1\u00d71 squares and fourteen dominoes. Total area = 3\u00d710 = 30. Two squares = 2, fourteen dominoes = 28, total 30. Good.\n\nWe need to count tilings. Let me think about placing 2 monominoes at some cells, and the remaining 28 cells must be tileable by dominoes (perfect matching).\n\nSo count = number of ways to choose 2 cells (positions of the two squares) such that the rest can be tiled by dominoes, weighted by... actually each choice of positions of squares gives a set of remaining cells; we need to count total tilings, which is sum over placements of squares of (number of domino tilings of remaining region).\n\nSo answer = sum over unordered pairs of cells {a,b} of (number of perfect matchings of the 3\u00d710 grid minus {a,b}).\n\nLet me use a coloring / matrix approach. This is essentially counting domino tilings of grid with 2 monomino holes.\n\nTotal = number of ways to tile 3\u00d710 with dominoes and 2 monominoes.\n\nThere's a nice generating approach. Let me think via transfer matrix on columns, tracking how many monominoes used.\n\nLet me set up a column-by-column DP. Grid 3 rows, 10 columns. We process columns left to right. State: which cells in current column boundary are already filled by horizontal dominoes protruding from previous column. We also count monominoes used (0,1,2).\n\nLet me define DP where we fill column by column. State = subset of 3 cells in the current column that are already filled (by horizontal domino from left). We decide how to fill remaining cells in the column using: vertical dominoes within column, horizontal dominoes extending to next column, or monominoes.\n\nLet me do a standard \"broken profile\" but column-based transfer.\n\nState entering a column: mask of cells already filled (from horizontal dominoes coming into this column). For each column we produce mask of cells that stick out to next column.\n\nLet me enumerate. Rows 0,1,2. In a column, cells not pre-filled must be covered by:\n- vertical domino (pair adjacent rows within column)\n- horizontal domino to the right (marks that cell filled in next column)\n- monomino (uses a monomino count).\n\nWe want to count arrangements with total monominoes = 2 over 10 columns, and final protrusion mask empty.\n\nLet me define transfer between (incoming mask) -> (outgoing mask, monominoes added).\n\nFor a given incoming mask `in` (cells already filled), the free cells are complement. We must cover all free cells. Cover options: choose a set of vertical dominoes among adjacent free cells, some cells go horizontal (to next), some become monominoes. Every free cell must be covered exactly once by one of these.\n\nLet me just enumerate configurations. Free cells = set F. Partition F into:\n- vertical domino pairs (adjacent rows, both in F)\n- horizontal (each such cell contributes to outgoing mask)\n- monomino cells.\n\nOutgoing mask = set of horizontal cells. Monomino count = number of monomino cells.\n\nWe need to enumerate all ways for each F.\n\nLet me just compute. Represent masks as bits {r0,r1,r2}.\n\nLet me use polynomial in x (x tracks monominoes). Transfer matrix T where T[in][out] = polynomial (sum over configs) x^{#monominoes}.\n\nSince we only care up to x^2, keep track.\n\nLet me enumerate all incoming masks (0..7) and for each, all ways to cover free cells.\n\nLet me define free cells set. I'll go through each incoming mask.\n\nI'll denote cells as bits: bit0=row0, bit1=row1, bit2=row2.\n\nFor a set of free cells, we choose a matching of vertical dominoes (adjacent rows 0-1 or 1-2), remaining free cells each independently either horizontal (out) or monomino.\n\nLet me enumerate.\n\nIncoming mask = m, free F = {0,1,2} \\ m.\n\nCase F = {} (m=7 full): outgoing = {}, monominoes 0. Only 1 way. Contributes: to out=0, x^0.\n\nLet me handle each F.\n\nI'll list all subsets F and possible outputs.\n\nDefine function: for free set F, enumerate vertical domino subsets (non-overlapping) that are subset of F, then remaining cells R = F minus vertical-covered. Each cell in R independently horizontal or monomino. Outgoing mask = set of horizontal cells among R. Monomino count = number chosen monomino.\n\nVertical domino options depend on F.\n\nLet me handle each F (F is complement of incoming mask). Since incoming mask ranges over all, F ranges over all subsets too.\n\nF = {} : one config, out={}, mono 0.\n\nF = {0} (single): no vertical possible. cell0: horizontal -> out={0}, mono0; or monomino -> out={}, mono1.\n\nF = {1}: cell1: out={1} mono0; or out={} mono1.\n\nF = {2}: out={2} mono0; or out={} mono1.\n\nF = {0,1}: vertical options: none, or domino(0,1).\n - no vertical: cells 0 and1 each H or M.\n   choices: both H -> out {0,1}, mono0\n   0H,1M -> out{0}, mono1\n   0M,1H -> out{1}, mono1\n   both M -> out{}, mono2\n - vertical domino(0,1): covers both, out{}, mono0.\n\nF = {1,2}: symmetric to {0,1}: vertical(1,2) possible.\n - no vertical: \n   both H -> out{1,2} mono0\n   1H2M -> out{1} mono1\n   1M2H -> out{2} mono1\n   both M -> out{} mono2\n - vertical(1,2): out{} mono0.\n\nF = {0,2}: not adjacent, no vertical.\n - 0H2H -> out{0,2} mono0\n - 0H2M -> out{0} mono1\n - 0M2H -> out{2} mono1\n - 0M2M -> out{} mono2\n\nF = {0,1,2}: vertical options: none, (0,1), (1,2). (can't have both (0,1)&(1,2) overlap at 1).\n - no vertical: cells 0,1,2 each H/M:\n   8 combos. out = set of H cells, mono = number of M.\n - vertical(0,1): remaining cell2: H-> out{2} mono0; M-> out{} mono1.\n - vertical(1,2): remaining cell0: H-> out{0} mono0; M-> out{} mono1.\n\nNow we need transfer matrix indexed by incoming mask (which = cells filled). Note incoming mask = m, F = complement. Outgoing mask becomes incoming mask for next column.\n\nWe start column1 with incoming mask = {} (nothing protruding from column 0). We process 10 columns. After column 10, outgoing mask must be {} (no protrusion beyond).\n\nWe want total monomino count = 2.\n\nLet me build transfer as function of incoming mask m -> generating over outgoing mask and mono.\n\nLet me tabulate T[m] = list of (out, mono).\n\nm=0 (F={0,1,2}):\n no vertical: for each of cells 0,1,2 choose H or M:\n  - all H: out=7({0,1,2}), mono0\n  - 0H1H2M: out={0,1}=3, mono1\n  - 0H1M2H: out={0,2}=5, mono1\n  - 0M1H2H: out={1,2}=6, mono1\n  - 0H1M2M: out={0}=1, mono2\n  - 0M1H2M: out={1}=2, mono2\n  - 0M1M2H: out={2}=4, mono2\n  - all M: out=0, mono3 (ignore, >2 but keep track maybe)\n vertical(0,1): cell2 H-> out={2}=4 mono0; M-> out=0 mono1\n vertical(1,2): cell0 H-> out={0}=1 mono0; M-> out=0 mono1\n\nm=1 (cell0 filled, F={1,2}):\n no vertical: \n  1H2H: out={1,2}=6 mono0\n  1H2M: out={1}=2 mono1\n  1M2H: out={2}=4 mono1\n  1M2M: out=0 mono2\n vertical(1,2): out=0 mono0\n\nm=2 (cell1 filled, F={0,2}):\n  0H2H: out={0,2}=5 mono0\n  0H2M: out={0}=1 mono1\n  0M2H: out={2}=4 mono1\n  0M2M: out=0 mono2\n\nm=3 (cells0,1 filled, F={2}):\n  2H: out={2}=4 mono0\n  2M: out=0 mono1\n\nm=4 (cell2 filled, F={0,1}):\n no vertical:\n  0H1H: out={0,1}=3 mono0\n  0H1M: out={0}=1 mono1\n  0M1H: out={1}=2 mono1\n  0M1M: out=0 mono2\n vertical(0,1): out=0 mono0\n\nm=5 (cells0,2 filled, F={1}):\n  1H: out={1}=2 mono0\n  1M: out=0 mono1\n\nm=6 (cells1,2 filled, F={0}):\n  0H: out={0}=1 mono0\n  0M: out=0 mono1\n\nm=7 (all filled, F={}):\n  out=0 mono0\n\nNow DP. Let state vector V[column][mask] = polynomial in monomino count (track counts for mono=0,1,2; ignore >2).\n\nStart before column1: incoming mask=0 with mono0 =1. Actually we begin processing column1 with incoming mask 0.\n\nLet me represent state as dict mask-> [c0,c1,c2] counts for total mono 0,1,2 so far.\n\nInitialize (before any column) : mask 0: [1,0,0].\n\nProcess 10 columns applying transfer T each time (transfer maps incoming mask distribution to outgoing mask distribution, adding mono).\n\nAfter 10 columns, answer = coefficient of mono=2 in mask 0.\n\nLet me compute step by step. This is tedious but let's do it.\n\nLet me denote state as vector over masks 0..7, each a triple.\n\nTransfer rule: for each incoming mask m with counts (a0,a1,a2), for each config (out, dm), add to new state at out: shift counts by dm.\n\nAdding polynomial: new[out] += counts shifted by dm (i.e., contributes to mono levels 0+dm etc), truncate at 2.\n\nLet me compute transitions as: newcounts[out][k] += oldcounts[m][k-dm].\n\nLet me first compute after column 1.\n\nInitial: only mask0 with [1,0,0].\n\nApply T[0] configs (with dm and out):\nList T[0]:\n (7,0),(3,1),(5,1),(6,1),(1,2),(2,2),(4,2),(0,3)[ignore level3],\n (4,0),(0,1),\n (1,0),(0,1)\n\nNow old mask0 counts [1,0,0] (only level0=1).\n\nFor each config, contributes level (0+dm) with value 1 to out.\n\nLet's accumulate:\n(7,0): out7 level0 +=1\n(3,1): out3 level1 +=1\n(5,1): out5 level1 +=1\n(6,1): out6 level1 +=1\n(1,2): out1 level2 +=1\n(2,2): out2 level2 +=1\n(4,2): out4 level2 +=1\n(0,3): ignore\n(4,0): out4 level0 +=1\n(0,1): out0 level1 +=1\n(1,0): out1 level0 +=1\n(0,1): out0 level1 +=1  (another)\n\nSo after column1 state:\nmask0: [0,2,0]  (level1=2 from two (0,1) configs)\nmask1: [1,0,1]  (level0=1 from (1,0); level2=1 from (1,2))\nmask2: [0,0,1]\nmask3: [0,1,0]\nmask4: [1,0,1]\nmask5: [0,1,0]\nmask6: [0,1,0]\nmask7: [1,0,0]\n\nLet me denote state S1:\nS1[0]=[0,2,0]\nS1[1]=[1,0,1]\nS1[2]=[0,0,1]\nS1[3]=[0,1,0]\nS1[4]=[1,0,1]\nS1[5]=[0,1,0]\nS1[6]=[0,1,0]\nS1[7]=[1,0,0]\n\nNow I need to apply transfer 9 more times. This is heavy. Let me organize transfer as linear operator. Since counts are triples, and dm shifts, we can treat as convolution with polynomial in x truncated at x^2.\n\nLet me express T as matrix M[m][out] = polynomial (in x). Then state polynomial vectors evolve: new[out] = sum_m old[m]*M[m][out], multiply polynomials truncated at x^2.\n\nLet me write M[m][out] polynomials (coefficient by x^dm).\n\nFrom T lists:\n\nm=0:\n out7: x^0 \u21921\n out3: x^1\n out5: x^1\n out6: x^1\n out1: x^2 + x^0 (from (1,2) and (1,0)) = 1 + x^2\n out2: x^2\n out4: x^2 + x^0 (from (4,2) and (4,0)) = 1 + x^2\n out0: x^1 + x^1 (two (0,1)) + x^3(ignore) = 2x\nSo M[0]:\n out0: 2x\n out1: 1 + x^2\n out2: x^2\n out3: x\n out4: 1 + x^2\n out5: x\n out6: x\n out7: 1\n\nm=1:\n out6: 1\n out2: x\n out4: x\n out0: x^2 (from 1M2M) + 1 (from vertical) = 1 + x^2\nSo M[1]:\n out0: 1 + x^2\n out2: x\n out4: x\n out6: 1\n\nm=2:\n out5: 1\n out1: x\n out4: x\n out0: x^2\nM[2]:\n out0: x^2\n out1: x\n out4: x\n out5: 1\n\nm=3:\n out4: 1\n out0: x\nM[3]:\n out0: x\n out4: 1\n\nm=4:\n out3: 1\n out1: x\n out2: x\n out0: x^2 (0M1M) + 1(vertical) = 1 + x^2\nM[4]:\n out0: 1 + x^2\n out1: x\n out2: x\n out3: 1\n\nm=5:\n out2: 1\n out0: x\nM[5]:\n out0: x\n out2: 1\n\nm=6:\n out1: 1\n out0: x\nM[6]:\n out0: x\n out1: 1\n\nm=7:\n out0: 1\nM[7]:\n out0: 1\n\nGood. Now evolve.\n\nWe need to apply this 10 times total starting from initial (mask0=1). We already did column1 (S1). Let's continue applying to reach S10, then read mask0 coefficient x^2.\n\nLet me define operation: given state vector S (8 polynomials), compute S' where S'[out] = sum_m S[m]*M[m][out] (truncate x^2).\n\nLet me write out formula for each out in terms of contributions from each m.\n\nWhich m contribute to each out? Let's invert:\n\nout0 receives from:\n m0: 2x\n m1: 1+x^2\n m2: x^2\n m3: x\n m4: 1+x^2\n m5: x\n m6: x\n m7: 1\n\nout1 receives from:\n m0: 1+x^2\n m2: x\n m4: x\n m6: 1\n\nout2 receives from:\n m0: x^2\n m1: x\n m4: x\n m5: 1\n\nout3 receives from:\n m0: x\n m4: 1\n\nout4 receives from:\n m0: 1+x^2\n m1: x\n m2: x\n m3: 1\n\nout5 receives from:\n m0: x\n m2: 1\n\nout6 receives from:\n m0: x\n m1: 1\n\nout7 receives from:\n m0: 1\n\nGreat, these are the update equations.\n\nNow let's iterate. Represent each polynomial as triple [c0,c1,c2].\n\nMultiplication by x shifts: [a,b,c]*x = [0,a,b] (drop c). By x^2: [a,b,c]*x^2=[0,0,a]. By (1+x^2): [a,b,c]*(1+x^2)=[a,b,c]+[0,0,a]=[a,b,c+a]. By 2x: [0,2a,2b].\n\nLet me define helper operations.\n\nLet me just compute S1 -> S2 -> ... \n\nS1:\n[0]=[0,2,0]\n[1]=[1,0,1]\n[2]=[0,0,1]\n[3]=[0,1,0]\n[4]=[1,0,1]\n[5]=[0,1,0]\n[6]=[0,1,0]\n[7]=[1,0,0]\n\nCompute S2.\n\nout0 = m0*2x + m1*(1+x^2) + m2*x^2 + m3*x + m4*(1+x^2) + m5*x + m6*x + m7*1\n\nm0=[0,2,0]; *2x = [0,0,4] (2x*[0,2,0]: shift and double: [0, 2*0,2*2]=[0,0,4]) yes.\nActually 2x*[c0,c1,c2] = [0,2c0,2c1] = [0,0,4]. Good.\n\nm1=[1,0,1]; *(1+x^2)=[1,0,1+1]=[1,0,2].\nm2=[0,0,1]; *x^2=[0,0,0].\nm3=[0,1,0]; *x=[0,0,1].\nm4=[1,0,1]; *(1+x^2)=[1,0,2].\nm5=[0,1,0]; *x=[0,0,1].\nm6=[0,1,0]; *x=[0,0,1].\nm7=[1,0,0]; *1=[1,0,0].\n\nSum out0: \nc0: 0+1+0+0+1+0+0+1=3\nc1: 0+0+0+0+0+0+0+0=0\nc2: 4+2+0+1+2+1+1+0=11\nout0=[3,0,11]\n\nout1 = m0*(1+x^2)+m2*x+m4*x+m6*1\nm0=[0,2,0]*(1+x^2)=[0,2,0]+[0,0,0]=[0,2,0] (since x^2*[0,2,0]=[0,0,0])\nm2=[0,0,1]*x=[0,0,0]\nm4=[1,0,1]*x=[0,1,0]\nm6=[0,1,0]*1=[0,1,0]\nSum: c0:0+0+0+0=0; c1:2+0+1+1=4; c2:0+0+0+0=0\nout1=[0,4,0]\n\nout2 = m0*x^2 + m1*x + m4*x + m5*1\nm0=[0,2,0]*x^2=[0,0,0]\nm1=[1,0,1]*x=[0,1,0]\nm4=[1,0,1]*x=[0,1,0]\nm5=[0,1,0]*1=[0,1,0]\nSum: c0:0;c1:0+1+1+1=3;c2:0\nout2=[0,3,0]\n\nout3 = m0*x + m4*1\nm0=[0,2,0]*x=[0,0,2]\nm4=[1,0,1]*1=[1,0,1]\nSum: [1,0,3]\nout3=[1,0,3]\n\nout4 = m0*(1+x^2)+m1*x+m2*x+m3*1\nm0=[0,2,0]*(1+x^2)=[0,2,0]\nm1=[1,0,1]*x=[0,1,0]\nm2=[0,0,1]*x=[0,0,0]\nm3=[0,1,0]*1=[0,1,0]\nSum: c0:0; c1:2+1+0+1=4; c2:0\nout4=[0,4,0]\n\nout5 = m0*x + m2*1\nm0=[0,2,0]*x=[0,0,2]\nm2=[0,0,1]*1=[0,0,1]\nSum=[0,0,3]\nout5=[0,0,3]\n\nout6 = m0*x + m1*1\nm0*x=[0,0,2]\nm1=[1,0,1]\nSum=[1,0,3]\nout6=[1,0,3]\n\nout7 = m0*1 = [0,2,0]\nout7=[0,2,0]\n\nS2:\n[0]=[3,0,11]\n[1]=[0,4,0]\n[2]=[0,3,0]\n[3]=[1,0,3]\n[4]=[0,4,0]\n[5]=[0,0,3]\n[6]=[1,0,3]\n[7]=[0,2,0]\n\nLet me sanity check total tilings count with 0 monominoes? Actually monominoes exactly 2 needed. Let's continue.\n\nHmm wait, but we should double check: with 3 columns processed the mask0 c0 might correspond to full domino tilings of 3\u00d7n? Actually a 3\u00d7odd rectangle can't be domino tiled (odd area). 3\u00d72 area 6 tileable. Let me not worry.\n\nLet me continue to S3.\n\nNow S2 values:\nm0=[3,0,11]\nm1=[0,4,0]\nm2=[0,3,0]\nm3=[1,0,3]\nm4=[0,4,0]\nm5=[0,0,3]\nm6=[1,0,3]\nm7=[0,2,0]\n\nout0 = m0*2x + m1*(1+x^2)+m2*x^2+m3*x+m4*(1+x^2)+m5*x+m6*x+m7*1\n\nm0*2x: 2x*[3,0,11]=[0,6,0] (=[0,2*3,2*0]=[0,6,0])\nm1*(1+x^2)=[0,4,0]+x^2*[0,4,0]=[0,4,0]+[0,0,0]=[0,4,0]\nm2*x^2=[0,3,0]*x^2=[0,0,0]\nm3*x=[1,0,3]*x=[0,1,0]\nm4*(1+x^2)=[0,4,0]+[0,0,0]=[0,4,0]\nm5*x=[0,0,3]*x=[0,0,0]\nm6*x=[1,0,3]*x=[0,1,0]\nm7*1=[0,2,0]\nSum c0: 0+0+0+0+0+0+0+0=0\nc1: 6+4+0+1+4+0+1+2=18\nc2: 0+0+0+0+0+0+0+0=0\nout0=[0,18,0]\n\nHmm c0=0. Interesting: 3\u00d73 has odd area with 2 monominoes leaves 7 cells odd\u2014can't be domino tiled, so mask0 requires even. After 3 columns area=9, minus mono. For mask0 to be reachable, everything covered; monomino count parity must match area parity. Area 9 odd, dominoes even coverage, so number of monominoes must be odd. So c0 (0 mono) and c2 (2 mono) should be 0, c1 (1 mono) possible. Indeed out0=[0,18,0]. Good consistency!\n\nGreat, parity check: after n columns, area=3n. Reachable mask0 has monomino count \u2261 3n mod 2 = n mod2. For n=10 (even), monomino count even, so 0 or 2 possible. Good.\n\nLet me continue computing.\n\nout1 = m0*(1+x^2)+m2*x+m4*x+m6*1\nm0*(1+x^2)=[3,0,11]+x^2*[3,0,11]=[3,0,11]+[0,0,3]=[3,0,14]\nm2*x=[0,3,0]*x=[0,0,3]\nm4*x=[0,4,0]*x=[0,0,4]\nm6*1=[1,0,3]\nSum: c0:3+0+0+1=4; c1:0+0+0+0=0; c2:14+3+4+3=24\nout1=[4,0,24]\n\nout2 = m0*x^2 + m1*x + m4*x + m5*1\nm0*x^2=[3,0,11]*x^2=[0,0,3]\nm1*x=[0,4,0]*x=[0,0,4]\nm4*x=[0,4,0]*x=[0,0,4]\nm5*1=[0,0,3]\nSum: c2:3+4+4+3=14; others 0\nout2=[0,0,14]\n\nout3 = m0*x + m4*1\nm0*x=[3,0,11]*x=[0,3,0]\nm4*1=[0,4,0]\nSum=[0,7,0]\nout3=[0,7,0]\n\nout4 = m0*(1+x^2)+m1*x+m2*x+m3*1\nm0*(1+x^2)=[3,0,14] (as before)\nm1*x=[0,4,0]*x=[0,0,4]\nm2*x=[0,3,0]*x=[0,0,3]\nm3*1=[1,0,3]\nSum: c0:3+0+0+1=4; c1:0; c2:14+4+3+3=24\nout4=[4,0,24]\n\nout5 = m0*x + m2*1\nm0*x=[0,3,0]\nm2*1=[0,3,0]\nSum=[0,6,0]\nout5=[0,6,0]\n\nout6 = m0*x + m1*1\nm0*x=[0,3,0]\nm1*1=[0,4,0]\nSum=[0,7,0]\nout6=[0,7,0]\n\nout7 = m0*1=[3,0,11]\nout7=[3,0,11]\n\nS3:\n[0]=[0,18,0]\n[1]=[4,0,24]\n[2]=[0,0,14]\n[3]=[0,7,0]\n[4]=[4,0,24]\n[5]=[0,6,0]\n[6]=[0,7,0]\n[7]=[3,0,11]\n\nParity check: after 3 columns, masks. For mask with certain filled cells, the covered area = 3n + |mask| roughly? Let's not worry; trust.\n\nContinue to S4.\n\nS3 values:\nm0=[0,18,0]\nm1=[4,0,24]\nm2=[0,0,14]\nm3=[0,7,0]\nm4=[4,0,24]\nm5=[0,6,0]\nm6=[0,7,0]\nm7=[3,0,11]\n\nout0 = m0*2x + m1*(1+x^2)+m2*x^2+m3*x+m4*(1+x^2)+m5*x+m6*x+m7*1\nm0*2x: 2x*[0,18,0]=[0,0,36]\nm1*(1+x^2)=[4,0,24]+x^2*[4,0,24]=[4,0,24]+[0,0,4]=[4,0,28]\nm2*x^2=[0,0,14]*x^2=[0,0,0]\nm3*x=[0,7,0]*x=[0,0,7]\nm4*(1+x^2)=[4,0,24]+[0,0,4]=[4,0,28]\nm5*x=[0,6,0]*x=[0,0,6]\nm6*x=[0,7,0]*x=[0,0,7]\nm7*1=[3,0,11]\nSum c0:0+4+0+0+4+0+0+3=11\nc1:0+0+0+0+0+0+0+0=0\nc2:36+28+0+7+28+6+7+11=123\nout0=[11,0,123]\n\nout1 = m0*(1+x^2)+m2*x+m4*x+m6*1\nm0*(1+x^2)=[0,18,0]+x^2*[0,18,0]=[0,18,0]+[0,0,0]=[0,18,0]\nm2*x=[0,0,14]*x=[0,0,0]\nm4*x=[4,0,24]*x=[0,4,0]\nm6*1=[0,7,0]\nSum: c0:0; c1:18+0+4+7=29; c2:0\nout1=[0,29,0]\n\nout2 = m0*x^2+m1*x+m4*x+m5*1\nm0*x^2=[0,18,0]*x^2=[0,0,0]\nm1*x=[4,0,24]*x=[0,4,0]\nm4*x=[4,0,24]*x=[0,4,0]\nm5*1=[0,6,0]\nSum: c1:0+4+4+6=14\nout2=[0,14,0]\n\nout3 = m0*x+m4*1\nm0*x=[0,18,0]*x=[0,0,18]\nm4*1=[4,0,24]\nSum=[4,0,42]\nout3=[4,0,42]\n\nout4 = m0*(1+x^2)+m1*x+m2*x+m3*1\nm0*(1+x^2)=[0,18,0]\nm1*x=[4,0,24]*x=[0,4,0]\nm2*x=[0,0,14]*x=[0,0,0]\nm3*1=[0,7,0]\nSum: c1:18+4+0+7=29\nout4=[0,29,0]\n\nout5 = m0*x+m2*1\nm0*x=[0,0,18]\nm2*1=[0,0,14]\nSum=[0,0,32]\nout5=[0,0,32]\n\nout6 = m0*x+m1*1\nm0*x=[0,0,18]\nm1*1=[4,0,24]\nSum=[4,0,42]\nout6=[4,0,42]\n\nout7 = m0*1=[0,18,0]\nout7=[0,18,0]\n\nS4:\n[0]=[11,0,123]\n[1]=[0,29,0]\n[2]=[0,14,0]\n[3]=[4,0,42]\n[4]=[0,29,0]\n[5]=[0,0,32]\n[6]=[4,0,42]\n[7]=[0,18,0]\n\nI notice a pattern: even columns look like S2 structure, odd like S3. Let me continue.\n\nCompute S5.\n\nS4 values:\nm0=[11,0,123]\nm1=[0,29,0]\nm2=[0,14,0]\nm3=[4,0,42]\nm4=[0,29,0]\nm5=[0,0,32]\nm6=[4,0,42]\nm7=[0,18,0]\n\nout0 = m0*2x + m1*(1+x^2)+m2*x^2+m3*x+m4*(1+x^2)+m5*x+m6*x+m7*1\nm0*2x=2x*[11,0,123]=[0,22,0]\nm1*(1+x^2)=[0,29,0]+x^2*[0,29,0]=[0,29,0]+[0,0,0]=[0,29,0]\nm2*x^2=[0,14,0]*x^2=[0,0,0]\nm3*x=[4,0,42]*x=[0,4,0]\nm4*(1+x^2)=[0,29,0]\nm5*x=[0,0,32]*x=[0,0,0]\nm6*x=[4,0,42]*x=[0,4,0]\nm7*1=[0,18,0]\nSum c0:0\nc1:22+29+0+4+29+0+4+18=106\nc2:0\nout0=[0,106,0]\n\nout1 = m0*(1+x^2)+m2*x+m4*x+m6*1\nm0*(1+x^2)=[11,0,123]+x^2*[11,0,123]=[11,0,123]+[0,0,11]=[11,0,134]\nm2*x=[0,14,0]*x=[0,0,14]\nm4*x=[0,29,0]*x=[0,0,29]\nm6*1=[4,0,42]\nSum: c0:11+0+0+4=15; c1:0; c2:134+14+29+42=219\nout1=[15,0,219]\n\nout2 = m0*x^2+m1*x+m4*x+m5*1\nm0*x^2=[11,0,123]*x^2=[0,0,11]\nm1*x=[0,29,0]*x=[0,0,29]\nm4*x=[0,29,0]*x=[0,0,29]\nm5*1=[0,0,32]\nSum c2:11+29+29+32=101\nout2=[0,0,101]\n\nout3 = m0*x+m4*1\nm0*x=[11,0,123]*x=[0,11,0]\nm4*1=[0,29,0]\nSum=[0,40,0]\nout3=[0,40,0]\n\nout4 = m0*(1+x^2)+m1*x+m2*x+m3*1\nm0*(1+x^2)=[11,0,134]\nm1*x=[0,29,0]*x=[0,0,29]\nm2*x=[0,14,0]*x=[0,0,14]\nm3*1=[4,0,42]\nSum: c0:11+0+0+4=15; c1:0; c2:134+29+14+42=219\nout4=[15,0,219]\n\nout5 = m0*x+m2*1\nm0*x=[0,11,0]\nm2*1=[0,14,0]\nSum=[0,25,0]\nout5=[0,25,0]\n\nout6 = m0*x+m1*1\nm0*x=[0,11,0]\nm1*1=[0,29,0]\nSum=[0,40,0]\nout6=[0,40,0]\n\nout7 = m0*1=[11,0,123]\nout7=[11,0,123]\n\nS5:\n[0]=[0,106,0]\n[1]=[15,0,219]\n[2]=[0,0,101]\n[3]=[0,40,0]\n[4]=[15,0,219]\n[5]=[0,25,0]\n[6]=[0,40,0]\n[7]=[11,0,123]\n\nContinue S6.\n\nS5 values:\nm0=[0,106,0]\nm1=[15,0,219]\nm2=[0,0,101]\nm3=[0,40,0]\nm4=[15,0,219]\nm5=[0,25,0]\nm6=[0,40,0]\nm7=[11,0,123]\n\nout0 = m0*2x + m1*(1+x^2)+m2*x^2+m3*x+m4*(1+x^2)+m5*x+m6*x+m7*1\nm0*2x=2x*[0,106,0]=[0,0,212]\nm1*(1+x^2)=[15,0,219]+[0,0,15]=[15,0,234]\nm2*x^2=[0,0,101]*x^2=[0,0,0]\nm3*x=[0,40,0]*x=[0,0,40]\nm4*(1+x^2)=[15,0,219]+[0,0,15]=[15,0,234]\nm5*x=[0,25,0]*x=[0,0,25]\nm6*x=[0,40,0]*x=[0,0,40]\nm7*1=[11,0,123]\nSum c0:0+15+0+0+15+0+0+11=41\nc1:0\nc2:212+234+0+40+234+25+40+123=908\nout0=[41,0,908]\n\nout1 = m0*(1+x^2)+m2*x+m4*x+m6*1\nm0*(1+x^2)=[0,106,0]+x^2*[0,106,0]=[0,106,0]\nm2*x=[0,0,101]*x=[0,0,0]\nm4*x=[15,0,219]*x=[0,15,0]\nm6*1=[0,40,0]\nSum: c1:106+0+15+40=161\nout1=[0,161,0]\n\nout2 = m0*x^2+m1*x+m4*x+m5*1\nm0*x^2=[0,106,0]*x^2=[0,0,0]\nm1*x=[15,0,219]*x=[0,15,0]\nm4*x=[15,0,219]*x=[0,15,0]\nm5*1=[0,25,0]\nSum c1:0+15+15+25=55\nout2=[0,55,0]\n\nout3 = m0*x+m4*1\nm0*x=[0,106,0]*x=[0,0,106]\nm4*1=[15,0,219]\nSum=[15,0,325]\nout3=[15,0,325]\n\nout4 = m0*(1+x^2)+m1*x+m2*x+m3*1\nm0*(1+x^2)=[0,106,0]\nm1*x=[15,0,219]*x=[0,15,0]\nm2*x=[0,0,101]*x=[0,0,0]\nm3*1=[0,40,0]\nSum c1:106+15+0+40=161\nout4=[0,161,0]\n\nout5 = m0*x+m2*1\nm0*x=[0,0,106]\nm2*1=[0,0,101]\nSum=[0,0,207]\nout5=[0,0,207]\n\nout6 = m0*x+m1*1\nm0*x=[0,0,106]\nm1*1=[15,0,219]\nSum=[15,0,325]\nout6=[15,0,325]\n\nout7 = m0*1=[0,106,0]\nout7=[0,106,0]\n\nS6:\n[0]=[41,0,908]\n[1]=[0,161,0]\n[2]=[0,55,0]\n[3]=[15,0,325]\n[4]=[0,161,0]\n[5]=[0,0,207]\n[6]=[15,0,325]\n[7]=[0,106,0]\n\nContinue S7.\n\nS6 values:\nm0=[41,0,908]\nm1=[0,161,0]\nm2=[0,55,0]\nm3=[15,0,325]\nm4=[0,161,0]\nm5=[0,0,207]\nm6=[15,0,325]\nm7=[0,106,0]\n\nout0 = m0*2x + m1*(1+x^2)+m2*x^2+m3*x+m4*(1+x^2)+m5*x+m6*x+m7*1\nm0*2x=2x*[41,0,908]=[0,82,0]\nm1*(1+x^2)=[0,161,0]+x^2*[0,161,0]=[0,161,0]\nm2*x^2=[0,55,0]*x^2=[0,0,0]\nm3*x=[15,0,325]*x=[0,15,0]\nm4*(1+x^2)=[0,161,0]\nm5*x=[0,0,207]*x=[0,0,0]\nm6*x=[15,0,325]*x=[0,15,0]\nm7*1=[0,106,0]\nSum c1:82+161+0+15+161+0+15+106=540\nout0=[0,540,0]\n\nout1 = m0*(1+x^2)+m2*x+m4*x+m6*1\nm0*(1+x^2)=[41,0,908]+[0,0,41]=[41,0,949]\nm2*x=[0,55,0]*x=[0,0,55]\nm4*x=[0,161,0]*x=[0,0,161]\nm6*1=[15,0,325]\nSum c0:41+0+0+15=56; c2:949+55+161+325=1490\nout1=[56,0,1490]\n\nout2 = m0*x^2+m1*x+m4*x+m5*1\nm0*x^2=[41,0,908]*x^2=[0,0,41]\nm1*x=[0,161,0]*x=[0,0,161]\nm4*x=[0,161,0]*x=[0,0,161]\nm5*1=[0,0,207]\nSum c2:41+161+161+207=570\nout2=[0,0,570]\n\nout3 = m0*x+m4*1\nm0*x=[41,0,908]*x=[0,41,0]\nm4*1=[0,161,0]\nSum=[0,202,0]\nout3=[0,202,0]\n\nout4 = m0*(1+x^2)+m1*x+m2*x+m3*1\nm0*(1+x^2)=[41,0,949]\nm1*x=[0,161,0]*x=[0,0,161]\nm2*x=[0,55,0]*x=[0,0,55]\nm3*1=[15,0,325]\nSum c0:41+0+0+15=56; c2:949+161+55+325=1490\nout4=[56,0,1490]\n\nout5 = m0*x+m2*1\nm0*x=[0,41,0]\nm2*1=[0,55,0]\nSum=[0,96,0]\nout5=[0,96,0]\n\nout6 = m0*x+m1*1\nm0*x=[0,41,0]\nm1*1=[0,161,0]\nSum=[0,202,0]\nout6=[0,202,0]\n\nout7 = m0*1=[41,0,908]\nout7=[41,0,908]\n\nS7:\n[0]=[0,540,0]\n[1]=[56,0,1490]\n[2]=[0,0,570]\n[3]=[0,202,0]\n[4]=[56,0,1490]\n[5]=[0,96,0]\n[6]=[0,202,0]\n[7]=[41,0,908]\n\nContinue S8.\n\nS7 values:\nm0=[0,540,0]\nm1=[56,0,1490]\nm2=[0,0,570]\nm3=[0,202,0]\nm4=[56,0,1490]\nm5=[0,96,0]\nm6=[0,202,0]\nm7=[41,0,908]\n\nout0 = m0*2x + m1*(1+x^2)+m2*x^2+m3*x+m4*(1+x^2)+m5*x+m6*x+m7*1\nm0*2x=2x*[0,540,0]=[0,0,1080]\nm1*(1+x^2)=[56,0,1490]+[0,0,56]=[56,0,1546]\nm2*x^2=[0,0,570]*x^2=[0,0,0]\nm3*x=[0,202,0]*x=[0,0,202]\nm4*(1+x^2)=[56,0,1490]+[0,0,56]=[56,0,1546]\nm5*x=[0,96,0]*x=[0,0,96]\nm6*x=[0,202,0]*x=[0,0,202]\nm7*1=[41,0,908]\nSum c0:0+56+0+0+56+0+0+41=153\nc2:1080+1546+0+202+1546+96+202+908=5580\nout0=[153,0,5580]\n\nLet me add c2: 1080+1546=2626; +202=2828; +1546=4374; +96=4470; +202=4672; +908=5580. Yes.\n\nout1 = m0*(1+x^2)+m2*x+m4*x+m6*1\nm0*(1+x^2)=[0,540,0]+x^2*[0,540,0]=[0,540,0]\nm2*x=[0,0,570]*x=[0,0,0]\nm4*x=[56,0,1490]*x=[0,56,0]\nm6*1=[0,202,0]\nSum c1:540+0+56+202=798\nout1=[0,798,0]\n\nout2 = m0*x^2+m1*x+m4*x+m5*1\nm0*x^2=[0,540,0]*x^2=[0,0,0]\nm1*x=[56,0,1490]*x=[0,56,0]\nm4*x=[56,0,1490]*x=[0,56,0]\nm5*1=[0,96,0]\nSum c1:0+56+56+96=208\nout2=[0,208,0]\n\nout3 = m0*x+m4*1\nm0*x=[0,540,0]*x=[0,0,540]\nm4*1=[56,0,1490]\nSum=[56,0,2030]\nout3=[56,0,2030]\n\nout4 = m0*(1+x^2)+m1*x+m2*x+m3*1\nm0*(1+x^2)=[0,540,0]\nm1*x=[56,0,1490]*x=[0,56,0]\nm2*x=[0,0,570]*x=[0,0,0]\nm3*1=[0,202,0]\nSum c1:540+56+0+202=798\nout4=[0,798,0]\n\nout5 = m0*x+m2*1\nm0*x=[0,0,540]\nm2*1=[0,0,570]\nSum=[0,0,1110]\nout5=[0,0,1110]\n\nout6 = m0*x+m1*1\nm0*x=[0,0,540]\nm1*1=[56,0,1490]\nSum=[56,0,2030]\nout6=[56,0,2030]\n\nout7 = m0*1=[0,540,0]\nout7=[0,540,0]\n\nS8:\n[0]=[153,0,5580]\n[1]=[0,798,0]\n[2]=[0,208,0]\n[3]=[56,0,2030]\n[4]=[0,798,0]\n[5]=[0,0,1110]\n[6]=[56,0,2030]\n[7]=[0,540,0]\n\nContinue S9.\n\nS8 values:\nm0=[153,0,5580]\nm1=[0,798,0]\nm2=[0,208,0]\nm3=[56,0,2030]\nm4=[0,798,0]\nm5=[0,0,1110]\nm6=[56,0,2030]\nm7=[0,540,0]\n\nout0 = m0*2x + m1*(1+x^2)+m2*x^2+m3*x+m4*(1+x^2)+m5*x+m6*x+m7*1\nm0*2x=2x*[153,0,5580]=[0,306,0]\nm1*(1+x^2)=[0,798,0]+x^2*[0,798,0]=[0,798,0]\nm2*x^2=[0,208,0]*x^2=[0,0,0]\nm3*x=[56,0,2030]*x=[0,56,0]\nm4*(1+x^2)=[0,798,0]\nm5*x=[0,0,1110]*x=[0,0,0]\nm6*x=[56,0,2030]*x=[0,56,0]\nm7*1=[0,540,0]\nSum c1:306+798+0+56+798+0+56+540=2554\nout0=[0,2554,0]\n\nout1 = m0*(1+x^2)+m2*x+m4*x+m6*1\nm0*(1+x^2)=[153,0,5580]+[0,0,153]=[153,0,5733]\nm2*x=[0,208,0]*x=[0,0,208]\nm4*x=[0,798,0]*x=[0,0,798]\nm6*1=[56,0,2030]\nSum c0:153+0+0+56=209; c2:5733+208+798+2030=8769\nout1=[209,0,8769]\n\nout2 = m0*x^2+m1*x+m4*x+m5*1\nm0*x^2=[153,0,5580]*x^2=[0,0,153]\nm1*x=[0,798,0]*x=[0,0,798]\nm4*x=[0,798,0]*x=[0,0,798]\nm5*1=[0,0,1110]\nSum c2:153+798+798+1110=2859\nout2=[0,0,2859]\n\nout3 = m0*x+m4*1\nm0*x=[153,0,5580]*x=[0,153,0]\nm4*1=[0,798,0]\nSum=[0,951,0]\nout3=[0,951,0]\n\nout4 = m0*(1+x^2)+m1*x+m2*x+m3*1\nm0*(1+x^2)=[153,0,5733]\nm1*x=[0,798,0]*x=[0,0,798]\nm2*x=[0,208,0]*x=[0,0,208]\nm3*1=[56,0,2030]\nSum c0:153+0+0+56=209; c2:5733+798+208+2030=8769\nout4=[209,0,8769]\n\nout5 = m0*x+m2*1\nm0*x=[0,153,0]\nm2*1=[0,208,0]\nSum=[0,361,0]\nout5=[0,361,0]\n\nout6 = m0*x+m1*1\nm0*x=[0,153,0]\nm1*1=[0,798,0]\nSum=[0,951,0]\nout6=[0,951,0]\n\nout7 = m0*1=[153,0,5580]\nout7=[153,0,5580]\n\nS9:\n[0]=[0,2554,0]\n[1]=[209,0,8769]\n[2]=[0,0,2859]\n[3]=[0,951,0]\n[4]=[209,0,8769]\n[5]=[0,361,0]\n[6]=[0,951,0]\n[7]=[153,0,5580]\n\nNow S10. We only need out0 final coefficient x^2.\n\nCompute out0 for S10 from S9.\n\nS9 values:\nm0=[0,2554,0]\nm1=[209,0,8769]\nm2=[0,0,2859]\nm3=[0,951,0]\nm4=[209,0,8769]\nm5=[0,361,0]\nm6=[0,951,0]\nm7=[153,0,5580]\n\nout0 = m0*2x + m1*(1+x^2)+m2*x^2+m3*x+m4*(1+x^2)+m5*x+m6*x+m7*1\n\nm0*2x=2x*[0,2554,0]=[0,0,5108]\nm1*(1+x^2)=[209,0,8769]+x^2*[209,0,8769]=[209,0,8769]+[0,0,209]=[209,0,8978]\nm2*x^2=[0,0,2859]*x^2=[0,0,0]\nm3*x=[0,951,0]*x=[0,0,951]\nm4*(1+x^2)=[209,0,8769]+[0,0,209]=[209,0,8978]\nm5*x=[0,361,0]*x=[0,0,361]\nm6*x=[0,951,0]*x=[0,0,951]\nm7*1=[153,0,5580]\n\nNow sum:\nc0: 0+209+0+0+209+0+0+153 = 571\nc1: 0\nc2: 5108 + 8978 + 0 + 951 + 8978 + 361 + 951 + 5580\n\nLet me sum c2:\n5108+8978 = 14086\n+951 = 15037\n+8978 = 24015\n+361 = 24376\n+951 = 25327\n+5580 = 30907\n\nout0=[571,0,30907]\n\nThe answer is coefficient of x^2 (exactly two monominoes) = 30907.\n\nWait, c0=571 corresponds to zero monominoes (full domino tiling of 3\u00d710)? But that would require zero squares; but we need exactly 2 squares. The value 571 is number of domino tilings of 3\u00d710. Let me check: known number of domino tilings of 3\u00d7n: for n=10 it's 571. Yes! 3\u00d710 domino tilings = 571. Great, that validates our transfer matrix.\n\nSo answer for exactly 2 monominoes = 30907.\n\nLet me double check the c2 sum and components once more.\n\nComponents of x^2 contributions:\nm0*2x: x^2 coeff = 2*(c1 of m0) = 2*2554 = 5108. \u2713\nm1*(1+x^2): x^2 coeff = c2(m1) + c0(m1) = 8769 + 209 = 8978. \u2713\nm2*x^2: c0(m2)=0 \u2192 0. \u2713\nm3*x: x^2 coeff = c1(m3) = 951. \u2713\nm4*(1+x^2): 8769+209=8978. \u2713\nm5*x: c1(m5)=361. \u2713\nm6*x: c1(m6)=951. \u2713\nm7*1: c2(m7)=5580. \u2713\n\nSum = 5108+8978+951+8978+361+951+5580 = 30907. \u2713\n\nAnswer: 30907.\n</think>\n\nUsing a column-by-column transfer matrix tracking the number of monomino (1\u00d71) tiles, I evolved the state over all 10 columns. As a sanity check, the coefficient for zero monominoes came out to 571, the known number of pure domino tilings of a 3\u00d710 rectangle \u2014 confirming the setup.\n\nThe coefficient for exactly two monominoes is:\n\n**30907**\n\nANSWER: 30907"
    },
    {
      "task_id": "PILOT-K1",
      "task_type": "code",
      "answer": "47206",
      "gold": "121090",
      "correct": false,
      "tokens_in": 171,
      "tokens_out": 32768,
      "cost_usd": 0.820055,
      "latency_s": 281.71,
      "timed_out": false,
      "solution": "<think>\nWe need to compute the sequence.\n\ns(1) = 2026.\nFor n >= 2: s(n) = s(n-1) + M(s(n-1)) * d(n), where M(k) is the largest decimal digit of s(n-1), d(n) is number of divisors of n.\n\nWe need s(2026). This requires iterating 2025 steps. Let me think if there's a pattern.\n\nM(k) is largest digit. Once numbers get large, likely max digit becomes 9 frequently. Let me think.\n\nActually we need to actually simulate. That's hard by hand for 2025 steps. Let me look for structure.\n\nNote M(s(n-1)) is between... s starts at 2026, max digit 6. Then increases.\n\nLet me consider: the increment each step is M * d(n). Sum of d(n) from n=2 to 2026... but M varies.\n\nHmm, maybe M becomes 9 quickly and stays 9. If M=9 always, then s(2026) = 2026 + 9 * sum_{n=2}^{2026} d(n).\n\nLet me check when M becomes 9 and whether it stays.\n\nLet me simulate the early steps.\n\ns(1)=2026, M=6.\nn=2: d(2)=2. s(2)=2026+6*2=2026+12=2038. M(2038)=8.\nn=3: d(3)=2. s=2038+8*2=2038+16=2054. M=5.\nn=4: d(4)=3. s=2054+5*3=2054+15=2069. M=9.\nn=5: d(5)=2. s=2069+9*2=2069+18=2087. M=8.\nn=6: d(6)=4. s=2087+8*4=2087+32=2119. M=9.\nn=7: d(7)=2. s=2119+9*2=2119+18=2137. M=7.\nn=8: d(8)=4. s=2137+7*4=2137+28=2165. M=6.\nn=9: d(9)=3. s=2165+6*3=2165+18=2183. M=8.\nn=10: d(10)=4. s=2183+8*4=2183+32=2215. M=5.\nn=11: d(11)=2. s=2215+5*2=2225. M=5.\nn=12: d(12)=6. s=2225+5*6=2225+30=2255. M=5.\nn=13: d=2. s=2255+5*2=2265. M=6.\nn=14: d=4. s=2265+6*4=2265+24=2289. M=9.\nn=15: d=4. s=2289+9*4=2289+36=2325. M=5.\nn=16: d=5. s=2325+5*5=2325+25=2350. M=5.\nn=17: d=2. s=2350+10=2360. M=6.\nn=18: d=6. s=2360+36=2396. M=9.\nn=19: d=2. s=2396+18=2414. M=4.\nn=20: d=6. s=2414+24=2438. M=8.\nn=21: d=4. s=2438+32=2470. M=7.\nn=22: d=4. s=2470+28=2498. M=9.\nn=23: d=2. s=2498+18=2516. M=6.\nn=24: d=8. s=2516+48=2564. M=6.\nn=25: d=3. s=2564+18=2582. M=8.\n\nThis is very tedious. M is not constant. Doing 2025 steps by hand is impractical but we must.\n\nLet me think about the magnitude. Each step adds roughly M*d(n). Average d(n) ~ ln(n) ~ 7.6 by n~2026. M averages maybe ~7-8. So each step adds ~50-60. Over 2025 steps that's ~100,000+. So s(2026) is around 100,000-130,000.\n\nHmm we really need to just simulate carefully. This is a lot but let me think whether there's a smarter observation.\n\nActually, this is a computational problem. Let me try to simulate but it's 2025 iterations... Not feasible manually with accuracy.\n\nLet me reconsider. Perhaps the answer requires actual computation. Let me try to be systematic and reduce error.\n\nAlternative: Maybe once s gets large enough that it always contains a 9 digit, M=9 constantly, so we only need sum of d(n).\n\nLet me think: after some point, does M stay 9? Not guaranteed since numbers can avoid having a 9. But probabilistically, over many steps M is often 9.\n\nWe need exact. Let me just carefully simulate. I'll need d(n) for n up to 2026. That's a huge amount.\n\nGiven the constraints, let me attempt a careful but efficient simulation, tracking s and computing d(n) for each n.\n\nThis is going to be extremely long. Let me think about whether the problem expects a computational answer that I should reproduce.\n\nGiven I must produce an integer, let me actually try to do the simulation but I'll batch.\n\nActually, let me reconsider - maybe there's a cleaner structure I'm missing. The key insight might be that M(s(n-1)) tends to be 9 for most large n, and we can compute sum of d(n).\n\nSum_{n=1}^{N} d(n) = sum_{k=1}^{N} floor(N/k). For N=2026:\nLet me compute D(2026) = sum_{k=1}^{2026} floor(2026/k).\n\nActually we need sum from n=2 to 2026 of d(n) = D(2026) - d(1) = D(2026)-1.\n\nApprox D(N) \u2248 N ln N + (2\u03b3-1)N \u2248 2026*7.614 + 0.1544*2026 \u2248 15426 + 313 = 15739. If all M=9, increment = 9*15739 \u2248 141650, s\u2248143676. But M isn't always 9.\n\nThis is genuinely a simulation problem. Without a computer, error-prone.\n\nLet me just commit to doing the simulation as carefully as I can. Actually, given the impracticality, let me reconsider the scale: 2025 steps. I'll try to do it but group.\n\nHmm, this is really a programming problem. Let me try to simulate mentally in chunks, being very careful. I'll track s(n) after each step.\n\nLet me restate: I have s values computed up to n=25: s(25)=2582.\n\nLet me continue. I need d(n).\n\nLet me list d(n) as I go.\n\nCurrent: s(25)=2582, M=8.\n\nn=26: d(26)=4. s=2582+8*4=2582+32=2614. M=6.\nn=27: d(27)=4. s=2614+6*4=2614+24=2638. M=8.\nn=28: d(28)=6. s=2638+8*6=2638+48=2686. M=8.\nn=29: d=2. s=2686+16=2702. M=7.\nn=30: d(30)=8. s=2702+7*8=2702+56=2758. M=8.\nn=31: d=2. s=2758+16=2774. M=7.\nn=32: d(32)=6. s=2774+7*6=2774+42=2816. M=8.\nn=33: d=4. s=2816+32=2848. M=8.\nn=34: d=4. s=2848+32=2880. M=8.\nn=35: d=4. s=2880+32=2912. M=9.\nn=36: d(36)=9. s=2912+9*9=2912+81=2993. M=9.\nn=37: d=2. s=2993+18=3011. M=3.\nn=38: d=4. s=3011+3*4=3011+12=3023. M=3.\nn=39: d=4. s=3023+12=3035. M=5.\nn=40: d(40)=8. s=3035+5*8=3035+40=3075. M=7.\nn=41: d=2. s=3075+14=3089. M=9.\nn=42: d(42)=8. s=3089+9*8=3089+72=3161. M=6.\nn=43: d=2. s=3161+12=3173. M=7.\nn=44: d=6. s=3173+7*6=3173+42=3215. M=5.\nn=45: d=6. s=3215+5*6=3215+30=3245. M=5.\nn=46: d=4. s=3245+20=3265. M=6.\nn=47: d=2. s=3265+12=3277. M=7.\nn=48: d(48)=10. s=3277+7*10=3277+70=3347. M=7.\nn=49: d=3. s=3347+21=3368. M=8.\nn=50: d(50)=6. s=3368+48=3416. M=6.\n\ns(50)=3416, M=6.\n\nThis is going to take forever for 2026. Let me think about the average behavior more and consider that maybe the intended answer is via computation.\n\nI worry about accuracy over 2000 steps. Let me reconsider whether the problem has a trick.\n\nActually maybe M(s(n-1)) \u2014 largest digit \u2014 after s exceeds ~a few thousand and grows, it's plausible M is often 9. But we need exact.\n\nI think I have to simulate. Given the enormous manual effort and error risk, let me try to be disciplined and continue, computing d(n) correctly.\n\nActually, let me reconsider the growth: by n=50, s=3416, increment ~48 over 25 steps... wait from s(25)=2582 to s(50)=3416, that's 834 over 25 steps, ~33/step. As d(n) grows, increments grow.\n\nTo reach n=2026 we need ~1976 more steps. Increments will grow to maybe ~60-70/step. Total maybe ~2582 + ... let's estimate final ~2000+ * ~average... Let me just estimate final around 100,000-130,000.\n\nI'll continue the simulation. This is laborious but let me push through in chunks of ~25, tracking carefully. I'll write d(n) values.\n\nLet me prepare d(n) for n=51..100:\n51=3*17: d=4\n52=4*13: d=6\n53 prime:2\n54=2*27=2*3^3: d=8\n55=5*11:4\n56=8*7:d=8\n57=3*19:4\n58=2*29:4\n59:2\n60=2^2*3*5: d=12\n61:2\n62=2*31:4\n63=7*9=3^2*7:6\n64=2^6:7\n65=5*13:4\n66=2*3*11:8\n67:2\n68=4*17:6\n69=3*23:4\n70=2*5*7:8\n71:2\n72=8*9=2^3*3^2:12\n73:2\n74=2*37:4\n75=3*25=3*5^2:6\n76=4*19:6\n77=7*11:4\n78=2*3*13:8\n79:2\n80=16*5=2^4*5:10\n81=3^4:5\n82=2*41:4\n83:2\n84=4*3*7=2^2*3*7:12\n85=5*17:4\n86=2*43:4\n87=3*29:4\n88=8*11:8\n89:2\n90=2*3^2*5:12\n91=7*13:4\n92=4*23:6\n93=3*31:4\n94=2*47:4\n95=5*19:4\n96=2^5*3:12\n97:2\n98=2*49:6\n99=9*11:6\n100=4*25:9\n\nContinue simulation from s(50)=3416, M=6.\n\nn=51: d=4. s=3416+6*4=3416+24=3440. M=4.\nn=52: d=6. s=3440+4*6=3440+24=3464. M=6.\nn=53: d=2. s=3464+12=3476. M=7.\nn=54: d=8. s=3476+7*8=3476+56=3532. M=5.\nn=55: d=4. s=3532+20=3552. M=5.\nn=56: d=8. s=3552+40=3592. M=9.\nn=57: d=4. s=3592+36=3628. M=8.\nn=58: d=4. s=3628+32=3660. M=6.\nn=59: d=2. s=3660+12=3672. M=7.\nn=60: d=12. s=3672+7*12=3672+84=3756. M=7.\nn=61: d=2. s=3756+14=3770. M=7.\nn=62: d=4. s=3770+28=3798. M=9.\nn=63: d=6. s=3798+54=3852. M=8.\nn=64: d=7. s=3852+56=3908. M=9.\nn=65: d=4. s=3908+36=3944. M=9.\nn=66: d=8. s=3944+72=4016. M=6.\nn=67: d=2. s=4016+12=4028. M=8.\nn=68: d=6. s=4028+48=4076. M=7.\nn=69: d=4. s=4076+28=4104. M=4.\nn=70: d=8. s=4104+32=4136. M=6.\nn=71: d=2. s=4136+12=4148. M=8.\nn=72: d=12. s=4148+96=4244. M=4.\nn=73: d=2. s=4244+8=4252. M=5.\nn=74: d=4. s=4252+20=4272. M=7.\nn=75: d=6. s=4272+42=4314. M=4.\nn=76: d=6. s=4314+24=4338. M=8.\nn=77: d=4. s=4338+32=4370. M=7.\nn=78: d=8. s=4370+56=4426. M=6.\nn=79: d=2. s=4426+12=4438. M=8.\nn=80: d=10. s=4438+80=4518. M=8.\nn=81: d=5. s=4518+40=4558. M=8.\nn=82: d=4. s=4558+32=4590. M=9.\nn=83: d=2. s=4590+18=4608. M=8.\nn=84: d=12. s=4608+96=4704. M=7.\nn=85: d=4. s=4704+28=4732. M=7.\nn=86: d=4. s=4732+28=4760. M=7.\nn=87: d=4. s=4760+28=4788. M=8.\nn=88: d=8. s=4788+64=4852. M=8.\nn=89: d=2. s=4852+16=4868. M=8.\nn=90: d=12. s=4868+96=4964. M=9.\nn=91: d=4. s=4964+36=5000. M=5.\nn=92: d=6. s=5000+30=5030. M=5.\nn=93: d=4. s=5030+20=5050. M=5.\nn=94: d=4. s=5050+20=5070. M=7.\nn=95: d=4. s=5070+28=5098. M=9.\nn=96: d=12. s=5098+108=5206. M=6.\nn=97: d=2. s=5206+12=5218. M=8.\nn=98: d=6. s=5218+48=5266. M=6.\nn=99: d=6. s=5266+36=5302. M=5.\nn=100: d=9. s=5302+45=5347. M=7.\n\ns(100)=5347, M=7.\n\nOK progress. 100 down, 1926 to go. This is extremely long. I'll keep going but I'm concerned about time. Let me continue steadily.\n\nd(n) for 101..150:\n101:2\n102=2*3*17:8\n103:2\n104=8*13:8\n105=3*5*7:8\n106=2*53:4\n107:2\n108=4*27=2^2*3^3:12\n109:2\n110=2*5*11:8\n111=3*37:4\n112=16*7=2^4*7:10\n113:2\n114=2*3*19:8\n115=5*23:4\n116=4*29:6\n117=9*13=3^2*13:6\n118=2*59:4\n119=7*17:4\n120=2^3*3*5:16\n121=11^2:3\n122=2*61:4\n123=3*41:4\n124=4*31:6\n125=5^3:4\n126=2*3^2*7:12\n127:2\n128=2^7:8\n129=3*43:4\n130=2*5*13:8\n131:2\n132=4*3*11=2^2*3*11:12\n133=7*19:4\n134=2*67:4\n135=27*5=3^3*5:8\n136=8*17:8\n137:2\n138=2*3*23:8\n139:2\n140=4*5*7=2^2*5*7:12\n141=3*47:4\n142=2*71:4\n143=11*13:4\n144=2^4*3^2:15\n145=5*29:4\n146=2*73:4\n147=3*49=3*7^2:6\n148=4*37:6\n149:2\n150=2*3*5^2:12\n\nContinue from s(100)=5347, M=7.\nn=101: d=2. s=5347+14=5361. M=6.\nn=102: d=8. s=5361+48=5409. M=9.\nn=103: d=2. s=5409+18=5427. M=7.\nn=104: d=8. s=5427+56=5483. M=8.\nn=105: d=8. s=5483+64=5547. M=7.\nn=106: d=4. s=5547+28=5575. M=7.\nn=107: d=2. s=5575+14=5589. M=9.\nn=108: d=12. s=5589+108=5697. M=9.\nn=109: d=2. s=5697+18=5715. M=7.\nn=110: d=8. s=5715+56=5771. M=7.\nn=111: d=4. s=5771+28=5799. M=9.\nn=112: d=10. s=5799+90=5889. M=9.\nn=113: d=2. s=5889+18=5907. M=9.\nn=114: d=8. s=5907+72=5979. M=9.\nn=115: d=4. s=5979+36=6015. M=6.\nn=116: d=6. s=6015+36=6051. M=6.\nn=117: d=6. s=6051+36=6087. M=8.\nn=118: d=4. s=6087+32=6119. M=9.\nn=119: d=4. s=6119+36=6155. M=6.\nn=120: d=16. s=6155+96=6251. M=6.\nn=121: d=3. s=6251+18=6269. M=9.\nn=122: d=4. s=6269+36=6305. M=6.\nn=123: d=4. s=6305+24=6329. M=9.\nn=124: d=6. s=6329+54=6383. M=8.\nn=125: d=4. s=6383+32=6415. M=6.\nn=126: d=12. s=6415+72=6487. M=8.\nn=127: d=2. s=6487+16=6503. M=6.\nn=128: d=8. s=6503+48=6551. M=6.\nn=129: d=4. s=6551+24=6575. M=7.\nn=130: d=8. s=6575+56=6631. M=6.\nn=131: d=2. s=6631+12=6643. M=6.\nn=132: d=12. s=6643+72=6715. M=7.\nn=133: d=4. s=6715+28=6743. M=7.\nn=134: d=4. s=6743+28=6771. M=7.\nn=135: d=8. s=6771+56=6827. M=8.\nn=136: d=8. s=6827+64=6891. M=9.\nn=137: d=2. s=6891+18=6909. M=9.\nn=138: d=8. s=6909+72=6981. M=9.\nn=139: d=2. s=6981+18=6999. M=9.\nn=140: d=12. s=6999+108=7107. M=7.\nn=141: d=4. s=7107+28=7135. M=7.\nn=142: d=4. s=7135+28=7163. M=7.\nn=143: d=4. s=7163+28=7191. M=9.\nn=144: d=15. s=7191+135=7326. M=7.\nn=145: d=4. s=7326+28=7354. M=7.\nn=146: d=4. s=7354+28=7382. M=8.\nn=147: d=6. s=7382+48=7430. M=7.\nn=148: d=6. s=7430+42=7472. M=7.\nn=149: d=2. s=7472+14=7486. M=8.\nn=150: d=12. s=7486+96=7582. M=8.\n\ns(150)=7582, M=8.\n\nContinuing. d(n) for 151..200:\n151:2\n152=8*19:8\n153=9*17=3^2*17:6\n154=2*7*11:8\n155=5*31:4\n156=4*3*13=2^2*3*13:12\n157:2\n158=2*79:4\n159=3*53:4\n160=2^5*5:12\n161=7*23:4\n162=2*3^4:10\n163:2\n164=4*41:6\n165=3*5*11:8\n166=2*83:4\n167:2\n168=2^3*3*7:16\n169=13^2:3\n170=2*5*17:8\n171=9*19=3^2*19:6\n172=4*43:6\n173:2\n174=2*3*29:8\n175=25*7=5^2*7:6\n176=16*11=2^4*11:10\n177=3*59:4\n178=2*89:4\n179:2\n180=2^2*3^2*5:18\n181:2\n182=2*7*13:8\n183=3*61:4\n184=8*23:8\n185=5*37:4\n186=2*3*31:8\n187=11*17:4\n188=4*47:6\n189=27*7=3^3*7:8\n190=2*5*19:8\n191:2\n192=2^6*3:14\n193:2\n194=2*97:4\n195=3*5*13:8\n196=4*49=2^2*7^2:9\n197:2\n198=2*9*11=2*3^2*11:12\n199:2\n200=2^3*5^2:12\n\nContinue from s(150)=7582, M=8.\nn=151: d=2. s=7582+16=7598. M=9.\nn=152: d=8. s=7598+72=7670. M=7.\nn=153: d=6. s=7670+42=7712. M=7.\nn=154: d=8. s=7712+56=7768. M=8.\nn=155: d=4. s=7768+32=7800. M=8.\nn=156: d=12. s=7800+96=7896. M=9.\nn=157: d=2. s=7896+18=7914. M=9.\nn=158: d=4. s=7914+36=7950. M=9.\nn=159: d=4. s=7950+36=7986. M=9.\nn=160: d=12. s=7986+108=8094. M=9.\nn=161: d=4. s=8094+36=8130. M=8.\nn=162: d=10. s=8130+80=8210. M=8.\nn=163: d=2. s=8210+16=8226. M=8.\nn=164: d=6. s=8226+48=8274. M=8.\nn=165: d=8. s=8274+64=8338. M=8.\nn=166: d=4. s=8338+32=8370. M=8.\nn=167: d=2. s=8370+16=8386. M=8.\nn=168: d=16. s=8386+128=8514. M=8.\nn=169: d=3. s=8514+24=8538. M=8.\nn=170: d=8. s=8538+64=8602. M=8.\nn=171: d=6. s=8602+48=8650. M=8.\nn=172: d=6. s=8650+48=8698. M=9.\nn=173: d=2. s=8698+18=8716. M=8.\nn=174: d=8. s=8716+64=8780. M=8.\nn=175: d=6. s=8780+48=8828. M=8.\nn=176: d=10. s=8828+80=8908. M=9.\nn=177: d=4. s=8908+36=8944. M=9.\nn=178: d=4. s=8944+36=8980. M=9.\nn=179: d=2. s=8980+18=8998. M=9.\nn=180: d=18. s=8998+162=9160. M=9.\nn=181: d=2. s=9160+18=9178. M=9.\nn=182: d=8. s=9178+72=9250. M=9.\nn=183: d=4. s=9250+36=9286. M=9.\nn=184: d=8. s=9286+72=9358. M=9.\nn=185: d=4. s=9358+36=9394. M=9.\nn=186: d=8. s=9394+72=9466. M=9.\nn=187: d=4. s=9466+36=9502. M=9.\nn=188: d=6. s=9502+54=9556. M=9.\nn=189: d=8. s=9556+72=9628. M=9.\nn=190: d=8. s=9628+72=9700. M=9.\nn=191: d=2. s=9700+18=9718. M=9.\nn=192: d=14. s=9718+126=9844. M=9.\nn=193: d=2. s=9844+18=9862. M=9.\nn=194: d=4. s=9862+36=9898. M=9.\nn=195: d=8. s=9898+72=9970. M=9.\nn=196: d=9. s=9970+81=10051. M=5.\nn=197: d=2. s=10051+10=10061. M=6.\nn=198: d=12. s=10061+72=10133. M=3.\nn=199: d=2. s=10133+6=10139. M=9.\nn=200: d=12. s=10139+108=10247. M=7.\n\ns(200)=10247, M=7.\n\nInteresting, we crossed 10000. Now numbers have 5 digits, higher chance of a 9.\n\nContinuing. This is halfway... no, only 200/2026 \u2248 10%. Ugh.\n\nLet me keep going but I realize the massive length. Let me persevere with focus.\n\nd(n) 201..250:\n201=3*67:4\n202=2*101:4\n203=7*29:4\n204=4*3*17=2^2*3*17:12\n205=5*41:4\n206=2*103:4\n207=9*23=3^2*23:6\n208=16*13=2^4*13:10\n209=11*19:4\n210=2*3*5*7:16\n211:2\n212=4*53:6\n213=3*71:4\n214=2*107:4\n215=5*43:4\n216=2^3*3^3:16\n217=7*31:4\n218=2*109:4\n219=3*73:4\n220=4*5*11=2^2*5*11:12\n221=13*17:4\n222=2*3*37:8\n223:2\n224=2^5*7:12\n225=9*25=3^2*5^2:9\n226=2*113:4\n227:2\n228=4*3*19=2^2*3*19:12\n229:2\n230=2*5*23:8\n231=3*7*11:8\n232=8*29:8\n233:2\n234=2*9*13=2*3^2*13:12\n235=5*47:4\n236=4*59:6\n237=3*79:4\n238=2*7*17:8\n239:2\n240=2^4*3*5:20\n241:2\n242=2*121=2*11^2:6\n243=3^5:6\n244=4*61:6\n245=5*49=5*7^2:6\n246=2*3*41:8\n247=13*19:4\n248=8*31:8\n249=3*83:4\n250=2*5^3:8\n\nContinue from s(200)=10247, M=7.\nn=201: d=4. s=10247+28=10275. M=7.\nn=202: d=4. s=10275+28=10303. M=3.\nn=203: d=4. s=10303+12=10315. M=5.\nn=204: d=12. s=10315+60=10375. M=7.\nn=205: d=4. s=10375+28=10403. M=4.\nn=206: d=4. s=10403+16=10419. M=9.\nn=207: d=6. s=10419+54=10473. M=7.\nn=208: d=10. s=10473+70=10543. M=5.\nn=209: d=4. s=10543+20=10563. M=6.\nn=210: d=16. s=10563+96=10659. M=9.\nn=211: d=2. s=10659+18=10677. M=7.\nn=212: d=6. s=10677+42=10719. M=9.\nn=213: d=4. s=10719+36=10755. M=7.\nn=214: d=4. s=10755+28=10783. M=8.\nn=215: d=4. s=10783+32=10815. M=8.\nn=216: d=16. s=10815+128=10943. M=9.\nn=217: d=4. s=10943+36=10979. M=9.\nn=218: d=4. s=10979+36=11015. M=5.\nn=219: d=4. s=11015+20=11035. M=5.\nn=220: d=12. s=11035+60=11095. M=9.\nn=221: d=4. s=11095+36=11131. M=3.\nn=222: d=8. s=11131+24=11155. M=5.\nn=223: d=2. s=11155+10=11165. M=6.\nn=224: d=12. s=11165+72=11237. M=7.\nn=225: d=9. s=11237+63=11300. M=3.\nn=226: d=4. s=11300+12=11312. M=3.\nn=227: d=2. s=11312+6=11318. M=8.\nn=228: d=12. s=11318+96=11414. M=4.\nn=229: d=2. s=11414+8=11422. M=4.\nn=230: d=8. s=11422+32=11454. M=5.\nn=231: d=8. s=11454+40=11494. M=9.\nn=232: d=8. s=11494+72=11566. M=6.\nn=233: d=2. s=11566+12=11578. M=8.\nn=234: d=12. s=11578+96=11674. M=7.\nn=235: d=4. s=11674+28=11702. M=7.\nn=236: d=6. s=11702+42=11744. M=7.\nn=237: d=4. s=11744+28=11772. M=7.\nn=238: d=8. s=11772+56=11828. M=8.\nn=239: d=2. s=11828+16=11844. M=8.\nn=240: d=20. s=11844+160=12004. M=4.\nn=241: d=2. s=12004+8=12012. M=2.\nn=242: d=6. s=12012+12=12024. M=4.\nn=243: d=6. s=12024+24=12048. M=8.\nn=244: d=6. s=12048+48=12096. M=9.\nn=245: d=6. s=12096+54=12150. M=5.\nn=246: d=8. s=12150+40=12190. M=9.\nn=247: d=4. s=12190+36=12226. M=6.\nn=248: d=8. s=12226+48=12274. M=7.\nn=249: d=4. s=12274+28=12302. M=3.\nn=250: d=8. s=12302+24=12326. M=6.\n\ns(250)=12326, M=6.\n\nThis is going to be an enormous computation. I've done 250 steps. Let me continue but I must accept the risk.\n\nLet me reconsider time budget - continuing all the way is ~8x more. Let me persevere efficiently.\n\nd(n) 251..300:\n251:2\n252=4*9*7=2^2*3^2*7:18\n253=11*23:4\n254=2*127:4\n255=3*5*17:8\n256=2^8:9\n257:2\n258=2*3*43:8\n259=7*37:4\n260=4*5*13=2^2*5*13:12\n261=9*29=3^2*29:6\n262=2*131:4\n263:2\n264=8*3*11=2^3*3*11:16\n265=5*53:4\n266=2*7*19:8\n267=3*89:4\n268=4*67:6\n269:2\n270=2*3^3*5:16\n271:2\n272=16*17=2^4*17:10\n273=3*7*13:8\n274=2*137:4\n275=25*11=5^2*11:6\n276=4*3*23=2^2*3*23:12\n277:2\n278=2*139:4\n279=9*31=3^2*31:6\n280=2^3*5*7:16\n281:2\n282=2*3*47:8\n283:2\n284=4*71:6\n285=3*5*19:8\n286=2*11*13:8\n287=7*41:4\n288=2^5*3^2:18\n289=17^2:3\n290=2*5*29:8\n291=3*97:4\n292=4*73:6\n293:2\n294=2*3*49=2*3*7^2:12\n295=5*59:4\n296=8*37:8\n297=27*11=3^3*11:8\n298=2*149:4\n299=13*23:4\n300=2^2*3*5^2:18\n\nContinue from s(250)=12326, M=6.\nn=251: d=2. s=12326+12=12338. M=8.\nn=252: d=18. s=12338+144=12482. M=8.\nn=253: d=4. s=12482+32=12514. M=5.\nn=254: d=4. s=12514+20=12534. M=5.\nn=255: d=8. s=12534+40=12574. M=7.\nn=256: d=9. s=12574+63=12637. M=7.\nn=257: d=2. s=12637+14=12651. M=6.\nn=258: d=8. s=12651+48=12699. M=9.\nn=259: d=4. s=12699+36=12735. M=7.\nn=260: d=12. s=12735+84=12819. M=9.\nn=261: d=6. s=12819+54=12873. M=8.\nn=262: d=4. s=12873+32=12905. M=9.\nn=263: d=2. s=12905+18=12923. M=9.\nn=264: d=16. s=12923+144=13067. M=7.\nn=265: d=4. s=13067+28=13095. M=9.\nn=266: d=8. s=13095+72=13167. M=7.\nn=267: d=4. s=13167+28=13195. M=9.\nn=268: d=6. s=13195+54=13249. M=9.\nn=269: d=2. s=13249+18=13267. M=7.\nn=270: d=16. s=13267+112=13379. M=9.\nn=271: d=2. s=13379+18=13397. M=9.\nn=272: d=10. s=13397+90=13487. M=8.\nn=273: d=8. s=13487+64=13551. M=5.\nn=274: d=4. s=13551+20=13571. M=7.\nn=275: d=6. s=13571+42=13613. M=6.\nn=276: d=12. s=13613+72=13685. M=8.\nn=277: d=2. s=13685+16=13701. M=7.\nn=278: d=4. s=13701+28=13729. M=9.\nn=279: d=6. s=13729+54=13783. M=8.\nn=280: d=16. s=13783+128=13911. M=9.\nn=281: d=2. s=13911+18=13929. M=9.\nn=282: d=8. s=13929+72=14001. M=4.\nn=283: d=2. s=14001+8=14009. M=9.\nn=284: d=6. s=14009+54=14063. M=6.\nn=285: d=8. s=14063+48=14111. M=4.\nn=286: d=8. s=14111+32=14143. M=4.\nn=287: d=4. s=14143+16=14159. M=9.\nn=288: d=18. s=14159+162=14321. M=4.\nn=289: d=3. s=14321+12=14333. M=4.\nn=290: d=8. s=14333+32=14365. M=6.\nn=291: d=4. s=14365+24=14389. M=9.\nn=292: d=6. s=14389+54=14443. M=4.\nn=293: d=2. s=14443+8=14451. M=5.\nn=294: d=12. s=14451+60=14511. M=5.\nn=295: d=4. s=14511+20=14531. M=5.\nn=296: d=8. s=14531+40=14571. M=7.\nn=297: d=8. s=14571+56=14627. M=7.\nn=298: d=4. s=14627+28=14655. M=6.\nn=299: d=4. s=14655+24=14679. M=9.\nn=300: d=18. s=14679+162=14841. M=8.\n\ns(300)=14841, M=8.\n\nContinuing. I'll persist.\n\nd(n) 301..350:\n301=7*43:4\n302=2*151:4\n303=3*101:4\n304=16*19=2^4*19:10\n305=5*61:4\n306=2*9*17=2*3^2*17:12\n307:2\n308=4*7*11=2^2*7*11:12\n309=3*103:4\n310=2*5*31:8\n311:2\n312=8*3*13=2^3*3*13:16\n313:2\n314=2*157:4\n315=9*5*7=3^2*5*7:12\n316=4*79:6\n317:2\n318=2*3*53:8\n319=11*29:4\n320=2^6*5:14\n321=3*107:4\n322=2*7*23:8\n323=17*19:4\n324=4*81=2^2*3^4:15\n325=25*13=5^2*13:6\n326=2*163:4\n327=3*109:4\n328=8*41:8\n329=7*47:4\n330=2*3*5*11:16\n331:2\n332=4*83:6\n333=9*37=3^2*37:6\n334=2*167:4\n335=5*67:4\n336=2^4*3*7:20\n337:2\n338=2*169=2*13^2:6\n339=3*113:4\n340=4*5*17=2^2*5*17:12\n341=11*31:4\n342=2*9*19=2*3^2*19:12\n343=7^3:4\n344=8*43:8\n345=3*5*23:8\n346=2*173:4\n347:2\n348=4*3*29=2^2*3*29:12\n349:2\n350=2*25*7=2*5^2*7:12\n\nContinue from s(300)=14841, M=8.\nn=301: d=4. s=14841+32=14873. M=8.\nn=302: d=4. s=14873+32=14905. M=9.\nn=303: d=4. s=14905+36=14941. M=9.\nn=304: d=10. s=14941+90=15031. M=5.\nn=305: d=4. s=15031+20=15051. M=5.\nn=306: d=12. s=15051+60=15111. M=5.\nn=307: d=2. s=15111+10=15121. M=5.\nn=308: d=12. s=15121+60=15181. M=8.\nn=309: d=4. s=15181+32=15213. M=5.\nn=310: d=8. s=15213+40=15253. M=5.\nn=311: d=2. s=15253+10=15263. M=6.\nn=312: d=16. s=15263+96=15359. M=9.\nn=313: d=2. s=15359+18=15377. M=7.\nn=314: d=4. s=15377+28=15405. M=5.\nn=315: d=12. s=15405+60=15465. M=6.\nn=316: d=6. s=15465+36=15501. M=5.\nn=317: d=2. s=15501+10=15511. M=5.\nn=318: d=8. s=15511+40=15551. M=5.\nn=319: d=4. s=15551+20=15571. M=7.\nn=320: d=14. s=15571+98=15669. M=9.\nn=321: d=4. s=15669+36=15705. M=7.\nn=322: d=8. s=15705+56=15761. M=7.\nn=323: d=4. s=15761+28=15789. M=9.\nn=324: d=15. s=15789+135=15924. M=9.\nn=325: d=6. s=15924+54=15978. M=9.\nn=326: d=4. s=15978+36=16014. M=6.\nn=327: d=4. s=16014+24=16038. M=8.\nn=328: d=8. s=16038+64=16102. M=6.\nn=329: d=4. s=16102+24=16126. M=6.\nn=330: d=16. s=16126+96=16222. M=6.\nn=331: d=2. s=16222+12=16234. M=6.\nn=332: d=6. s=16234+36=16270. M=7.\nn=333: d=6. s=16270+42=16312. M=6.\nn=334: d=4. s=16312+24=16336. M=6.\nn=335: d=4. s=16336+24=16360. M=6.\nn=336: d=20. s=16360+120=16480. M=8.\nn=337: d=2. s=16480+16=16496. M=9.\nn=338: d=6. s=16496+54=16550. M=6.\nn=339: d=4. s=16550+24=16574. M=7.\nn=340: d=12. s=16574+84=16658. M=8.\nn=341: d=4. s=16658+32=16690. M=9.\nn=342: d=12. s=16690+108=16798. M=9.\nn=343: d=4. s=16798+36=16834. M=8.\nn=344: d=8. s=16834+64=16898. M=9.\nn=345: d=8. s=16898+72=16970. M=9.\nn=346: d=4. s=16970+36=17006. M=7.\nn=347: d=2. s=17006+14=17020. M=7.\nn=348: d=12. s=17020+84=17104. M=7.\nn=349: d=2. s=17104+14=17118. M=8.\nn=350: d=12. s=17118+96=17214. M=7.\n\ns(350)=17214, M=7.\n\nContinue. d(n) 351..400:\n351=27*13=3^3*13:8\n352=2^5*11:12\n353:2\n354=2*3*59:8\n355=5*71:4\n356=4*89:6\n357=3*7*17:8\n358=2*179:4\n359:2\n360=2^3*3^2*5:24\n361=19^2:3\n362=2*181:4\n363=3*121=3*11^2:6\n364=4*7*13=2^2*7*13:12\n365=5*73:4\n366=2*3*61:8\n367:2\n368=16*23=2^4*23:10\n369=9*41=3^2*41:6\n370=2*5*37:8\n371=7*53:4\n372=4*3*31=2^2*3*31:12\n373:2\n374=2*11*17:8\n375=3*125=3*5^3:8\n376=8*47:8\n377=13*29:4\n378=2*27*7=2*3^3*7:16\n379:2\n380=4*5*19=2^2*5*19:12\n381=3*127:4\n382=2*191:4\n383:2\n384=2^7*3:16\n385=5*7*11:8\n386=2*193:4\n387=9*43=3^2*43:6\n388=4*97:6\n389:2\n390=2*3*5*13:16\n391=17*23:4\n392=8*49=2^3*7^2:12\n393=3*131:4\n394=2*197:4\n395=5*79:4\n396=4*9*11=2^2*3^2*11:18\n397:2\n398=2*199:4\n399=3*7*19:8\n400=2^4*5^2:15\n\nContinue from s(350)=17214, M=7.\nn=351: d=8. s=17214+56=17270. M=7.\nn=352: d=12. s=17270+84=17354. M=7.\nn=353: d=2. s=17354+14=17368. M=8.\nn=354: d=8. s=17368+64=17432. M=7.\nn=355: d=4. s=17432+28=17460. M=7.\nn=356: d=6. s=17460+42=17502. M=7.\nn=357: d=8. s=17502+56=17558. M=8.\nn=358: d=4. s=17558+32=17590. M=9.\nn=359: d=2. s=17590+18=17608. M=8.\nn=360: d=24. s=17608+192=17800. M=8.\nn=361: d=3. s=17800+24=17824. M=8.\nn=362: d=4. s=17824+32=17856. M=8.\nn=363: d=6. s=17856+48=17904. M=9.\nn=364: d=12. s=17904+108=18012. M=8.\nn=365: d=4. s=18012+32=18044. M=8.\nn=366: d=8. s=18044+64=18108. M=8.\nn=367: d=2. s=18108+16=18124. M=8.\nn=368: d=10. s=18124+80=18204. M=8.\nn=369: d=6. s=18204+48=18252. M=8.\nn=370: d=8. s=18252+64=18316. M=8.\nn=371: d=4. s=18316+32=18348. M=8.\nn=372: d=12. s=18348+96=18444. M=8.\nn=373: d=2. s=18444+16=18460. M=8.\nn=374: d=8. s=18460+64=18524. M=8.\nn=375: d=8. s=18524+64=18588. M=8.\nn=376: d=8. s=18588+64=18652. M=8.\nn=377: d=4. s=18652+32=18684. M=8.\nn=378: d=16. s=18684+128=18812. M=8.\nn=379: d=2. s=18812+16=18828. M=8.\nn=380: d=12. s=18828+96=18924. M=9.\nn=381: d=4. s=18924+36=18960. M=9.\nn=382: d=4. s=18960+36=18996. M=9.\nn=383: d=2. s=18996+18=19014. M=9.\nn=384: d=16. s=19014+144=19158. M=9.\nn=385: d=8. s=19158+72=19230. M=9.\nn=386: d=4. s=19230+36=19266. M=9.\nn=387: d=6. s=19266+54=19320. M=9.\nn=388: d=6. s=19320+54=19374. M=9.\nn=389: d=2. s=19374+18=19392. M=9.\nn=390: d=16. s=19392+144=19536. M=9.\nn=391: d=4. s=19536+36=19572. M=9.\nn=392: d=12. s=19572+108=19680. M=9.\nn=393: d=4. s=19680+36=19716. M=9.\nn=394: d=4. s=19716+36=19752. M=9.\nn=395: d=4. s=19752+36=19788. M=9.\nn=396: d=18. s=19788+162=19950. M=9.\nn=397: d=2. s=19950+18=19968. M=9.\nn=398: d=4. s=19968+36=20004. M=4.\nn=399: d=8. s=20004+32=20036. M=6.\nn=400: d=15. s=20036+90=20126. M=6.\n\ns(400)=20126, M=6.\n\nContinue. d(n) 401..450:\n401:2\n402=2*3*67:8\n403=13*31:4\n404=4*101:6\n405=81*5=3^4*5:10\n406=2*7*29:8\n407=11*37:4\n408=8*3*17=2^3*3*17:16\n409:2\n410=2*5*41:8\n411=3*137:4\n412=4*103:6\n413=7*59:4\n414=2*9*23=2*3^2*23:12\n415=5*83:4\n416=2^5*13:12\n417=3*139:4\n418=2*11*19:8\n419:2\n420=2^2*3*5*7:24\n421:2\n422=2*211:4\n423=9*47=3^2*47:6\n424=8*53:8\n425=25*17=5^2*17:6\n426=2*3*71:8\n427=7*61:4\n428=4*107:6\n429=3*11*13:8\n430=2*5*43:8\n431:2\n432=2^4*27=2^4*3^3:20\n433:2\n434=2*7*31:8\n435=3*5*29:8\n436=4*109:6\n437=19*23:4\n438=2*3*73:8\n439:2\n440=8*5*11=2^3*5*11:16\n441=9*49=3^2*7^2:9\n442=2*13*17:8\n443:2\n444=4*3*37=2^2*3*37:12\n445=5*89:4\n446=2*223:4\n447=3*149:4\n448=2^6*7:14\n449:2\n450=2*9*25=2*3^2*5^2:18\n\nContinue from s(400)=20126, M=6.\nn=401: d=2. s=20126+12=20138. M=8.\nn=402: d=8. s=20138+64=20202. M=2.\nn=403: d=4. s=20202+8=20210. M=2.\nn=404: d=6. s=20210+12=20222. M=2.\nn=405: d=10. s=20222+20=20242. M=4.\nn=406: d=8. s=20242+32=20274. M=7.\nn=407: d=4. s=20274+28=20302. M=3.\nn=408: d=16. s=20302+48=20350. M=5.\nn=409: d=2. s=20350+10=20360. M=6.\nn=410: d=8. s=20360+48=20408. M=8.\nn=411: d=4. s=20408+32=20440. M=4.\nn=412: d=6. s=20440+24=20464. M=6.\nn=413: d=4. s=20464+24=20488. M=8.\nn=414: d=12. s=20488+96=20584. M=8.\nn=415: d=4. s=20584+32=20616. M=6.\nn=416: d=12. s=20616+72=20688. M=8.\nn=417: d=4. s=20688+32=20720. M=7.\nn=418: d=8. s=20720+56=20776. M=7.\nn=419: d=2. s=20776+14=20790. M=9.\nn=420: d=24. s=20790+216=21006. M=6.\nn=421: d=2. s=21006+12=21018. M=8.\nn=422: d=4. s=21018+32=21050. M=5.\nn=423: d=6. s=21050+30=21080. M=8.\nn=424: d=8. s=21080+64=21144. M=4.\nn=425: d=6. s=21144+24=21168. M=8.\nn=426: d=8. s=21168+64=21232. M=3.\nn=427: d=4. s=21232+12=21244. M=4.\nn=428: d=6. s=21244+24=21268. M=8.\nn=429: d=8. s=21268+64=21332. M=3.\nn=430: d=8. s=21332+24=21356. M=6.\nn=431: d=2. s=21356+12=21368. M=8.\nn=432: d=20. s=21368+160=21528. M=8.\nn=433: d=2. s=21528+16=21544. M=5.\nn=434: d=8. s=21544+40=21584. M=8.\nn=435: d=8. s=21584+64=21648. M=8.\nn=436: d=6. s=21648+48=21696. M=9.\nn=437: d=4. s=21696+36=21732. M=7.\nn=438: d=8. s=21732+56=21788. M=8.\nn=439: d=2. s=21788+16=21804. M=8.\nn=440: d=16. s=21804+128=21932. M=9.\nn=441: d=9. s=21932+81=22013. M=3.\nn=442: d=8. s=22013+24=22037. M=7.\nn=443: d=2. s=22037+14=22051. M=5.\nn=444: d=12. s=22051+60=22111. M=2.\nn=445: d=4. s=22111+8=22119. M=9.\nn=446: d=4. s=22119+36=22155. M=5.\nn=447: d=4. s=22155+20=22175. M=7.\nn=448: d=14. s=22175+98=22273. M=7.\nn=449: d=2. s=22273+14=22287. M=8.\nn=450: d=18. s=22287+144=22431. M=4.\n\ns(450)=22431, M=4.\n\nContinue. d(n) 451..500:\n451=11*41:4\n452=4*113:6\n453=3*151:4\n454=2*227:4\n455=5*7*13:8\n456=8*3*19=2^3*3*19:16\n457:2\n458=2*229:4\n459=27*17=3^3*17:8\n460=4*5*23=2^2*5*23:12\n461:2\n462=2*3*7*11:16\n463:2\n464=16*29=2^4*29:10\n465=3*5*31:8\n466=2*233:4\n467:2\n468=4*9*13=2^2*3^2*13:18\n469=7*67:4\n470=2*5*47:8\n471=3*157:4\n472=8*59:8\n473=11*43:4\n474=2*3*79:8\n475=25*19=5^2*19:6\n476=4*7*17=2^2*7*17:12\n477=9*53=3^2*53:6\n478=2*239:4\n479:2\n480=2^5*3*5:24\n481=13*37:4\n482=2*241:4\n483=3*7*23:8\n484=4*121=2^2*11^2:9\n485=5*97:4\n486=2*3^5:12\n487:2\n488=8*61:8\n489=3*163:4\n490=2*5*49=2*5*7^2:12\n491:2\n492=4*3*41=2^2*3*41:12\n493=17*29:4\n494=2*13*19:8\n495=9*5*11=3^2*5*11:12\n496=16*31=2^4*31:10\n497=7*71:4\n498=2*3*83:8\n499:2\n500=4*125=2^2*5^3:12\n\nContinue from s(450)=22431, M=4.\nn=451: d=4. s=22431+16=22447. M=7.\nn=452: d=6. s=22447+42=22489. M=9.\nn=453: d=4. s=22489+36=22525. M=5.\nn=454: d=4. s=22525+20=22545. M=5.\nn=455: d=8. s=22545+40=22585. M=8.\nn=456: d=16. s=22585+128=22713. M=7.\nn=457: d=2. s=22713+14=22727. M=7.\nn=458: d=4. s=22727+28=22755. M=7.\nn=459: d=8. s=22755+56=22811. M=8.\nn=460: d=12. s=22811+96=22907. M=9.\nn=461: d=2. s=22907+18=22925. M=9.\nn=462: d=16. s=22925+144=23069. M=9.\nn=463: d=2. s=23069+18=23087. M=8.\nn=464: d=10. s=23087+80=23167. M=7.\nn=465: d=8. s=23167+56=23223. M=3.\nn=466: d=4. s=23223+12=23235. M=5.\nn=467: d=2. s=23235+10=23245. M=5.\nn=468: d=18. s=23245+90=23335. M=5.\nn=469: d=4. s=23335+20=23355. M=5.\nn=470: d=8. s=23355+40=23395. M=9.\nn=471: d=4. s=23395+36=23431. M=4.\nn=472: d=8. s=23431+32=23463. M=6.\nn=473: d=4. s=23463+24=23487. M=8.\nn=474: d=8. s=23487+64=23551. M=5.\nn=475: d=6. s=23551+30=23581. M=8.\nn=476: d=12. s=23581+96=23677. M=7.\nn=477: d=6. s=23677+42=23719. M=9.\nn=478: d=4. s=23719+36=23755. M=7.\nn=479: d=2. s=23755+14=23769. M=9.\nn=480: d=24. s=23769+216=23985. M=9.\nn=481: d=4. s=23985+36=24021. M=4.\nn=482: d=4. s=24021+16=24037. M=7.\nn=483: d=8. s=24037+56=24093. M=9.\nn=484: d=9. s=24093+81=24174. M=7.\nn=485: d=4. s=24174+28=24202. M=4.\nn=486: d=12. s=24202+48=24250. M=5.\nn=487: d=2. s=24250+10=24260. M=6.\nn=488: d=8. s=24260+48=24308. M=8.\nn=489: d=4. s=24308+32=24340. M=4.\nn=490: d=12. s=24340+48=24388. M=8.\nn=491: d=2. s=24388+16=24404. M=4.\nn=492: d=12. s=24404+48=24452. M=5.\nn=493: d=4. s=24452+20=24472. M=7.\nn=494: d=8. s=24472+56=24528. M=8.\nn=495: d=12. s=24528+96=24624. M=6.\nn=496: d=10. s=24624+60=24684. M=8.\nn=497: d=4. s=24684+32=24716. M=7.\nn=498: d=8. s=24716+56=24772. M=7.\nn=499: d=2. s=24772+14=24786. M=8.\nn=500: d=12. s=24786+96=24882. M=8.\n\ns(500)=24882, M=8.\n\nQuarter done. Numbers ~24882 at n=500. Roughly linear ~ let's see growth: from s(400)=20126 to s(500)=24882, delta 4756 over 100 steps, ~47.6/step. Continuing similarly, each 100 adds ~5000. So by n=2026, s \u2248 24882 + 1526*~50 \u2248 24882 + 76000 \u2248 100,000+. But increments grow with d(n).\n\nI must continue. Let me push on.\n\nd(n) 501..550:\n501=3*167:4\n502=2*251:4\n503:2\n504=2^3*3^2*7:24\n505=5*101:4\n506=2*11*23:8\n507=3*169=3*13^2:6\n508=4*127:6\n509:2\n510=2*3*5*17:16\n511=7*73:4\n512=2^9:10\n513=27*19=3^3*19:8\n514=2*257:4\n515=5*103:4\n516=4*3*43=2^2*3*43:12\n517=11*47:4\n518=2*7*37:8\n519=3*173:4\n520=8*5*13=2^3*5*13:16\n521:2\n522=2*9*29=2*3^2*29:12\n523:2\n524=4*131:6\n525=3*25*7=3*5^2*7:12\n526=2*263:4\n527=17*31:4\n528=2^4*3*11:20\n529=23^2:3\n530=2*5*53:8\n531=9*59=3^2*59:6\n532=4*7*19=2^2*7*19:12\n533=13*41:4\n534=2*3*89:8\n535=5*107:4\n536=8*67:8\n537=3*179:4\n538=2*269:4\n539=7^2*11:6\n540=2^2*3^3*5:24\n541:2\n542=2*271:4\n543=3*181:4\n544=2^5*17:12\n545=5*109:4\n546=2*3*7*13:16\n547:2\n548=4*137:6\n549=9*61=3^2*61:6\n550=2*25*11=2*5^2*11:12\n\nContinue from s(500)=24882, M=8.\nn=501: d=4. s=24882+32=24914. M=9.\nn=502: d=4. s=24914+36=24950. M=9.\nn=503: d=2. s=24950+18=24968. M=9.\nn=504: d=24. s=24968+216=25184. M=8.\nn=505: d=4. s=25184+32=25216. M=6.\nn=506: d=8. s=25216+48=25264. M=6.\nn=507: d=6. s=25264+36=25300. M=5.\nn=508: d=6. s=25300+30=25330. M=5.\nn=509: d=2. s=25330+10=25340. M=5.\nn=510: d=16. s=25340+80=25420. M=5.\nn=511: d=4. s=25420+20=25440. M=5.\nn=512: d=10. s=25440+50=25490. M=9.\nn=513: d=8. s=25490+72=25562. M=6.\nn=514: d=4. s=25562+24=25586. M=8.\nn=515: d=4. s=25586+32=25618. M=8.\nn=516: d=12. s=25618+96=25714. M=7.\nn=517: d=4. s=25714+28=25742. M=7.\nn=518: d=8. s=25742+56=25798. M=9.\nn=519: d=4. s=25798+36=25834. M=8.\nn=520: d=16. s=25834+128=25962. M=9.\nn=521: d=2. s=25962+18=25980. M=9.\nn=522: d=12. s=25980+108=26088. M=8.\nn=523: d=2. s=26088+16=26104. M=6.\nn=524: d=6. s=26104+36=26140. M=6.\nn=525: d=12. s=26140+72=26212. M=6.\nn=526: d=4. s=26212+24=26236. M=6.\nn=527: d=4. s=26236+24=26260. M=6.\nn=528: d=20. s=26260+120=26380. M=8.\nn=529: d=3. s=26380+24=26404. M=6.\nn=530: d=8. s=26404+48=26452. M=6.\nn=531: d=6. s=26452+36=26488. M=8.\nn=532: d=12. s=26488+96=26584. M=8.\nn=533: d=4. s=26584+32=26616. M=6.\nn=534: d=8. s=26616+48=26664. M=6.\nn=535: d=4. s=26664+24=26688. M=8.\nn=536: d=8. s=26688+64=26752. M=7.\nn=537: d=4. s=26752+28=26780. M=8.\nn=538: d=4. s=26780+32=26812. M=8.\nn=539: d=6. s=26812+48=26860. M=8.\nn=540: d=24. s=26860+192=27052. M=7.\nn=541: d=2. s=27052+14=27066. M=7.\nn=542: d=4. s=27066+28=27094. M=9.\nn=543: d=4. s=27094+36=27130. M=7.\nn=544: d=12. s=27130+84=27214. M=7.\nn=545: d=4. s=27214+28=27242. M=7.\nn=546: d=16. s=27242+112=27354. M=7.\nn=547: d=2. s=27354+14=27368. M=8.\nn=548: d=6. s=27368+48=27416. M=7.\nn=549: d=6. s=27416+42=27458. M=8.\nn=550: d=12. s=27458+96=27554. M=7.\n\ns(550)=27554, M=7.\n\nContinue. d(n) 551..600:\n551=19*29:4\n552=8*3*23=2^3*3*23:16\n553=7*79:4\n554=2*277:4\n555=3*5*37:8\n556=4*139:6\n557:2\n558=2*9*31=2*3^2*31:12\n559=13*43:4\n560=2^4*5*7:20\n561=3*11*17:8\n562=2*281:4\n563:2\n564=4*3*47=2^2*3*47:12\n565=5*113:4\n566=2*283:4\n567=81*7=3^4*7:10\n568=8*71:8\n569:2\n570=2*3*5*19:16\n571:2\n572=4*11*13=2^2*11*13:12\n573=3*191:4\n574=2*7*41:8\n575=25*23=5^2*23:6\n576=2^6*3^2:21\n577:2\n578=2*17^2:6\n579=3*193:4\n580=4*5*29=2^2*5*29:12\n581=7*83:4\n582=2*3*97:8\n583=11*53:4\n584=8*73:8\n585=9*5*13=3^2*5*13:12\n586=2*293:4\n587:2\n588=4*3*49=2^2*3*7^2:18\n589=19*31:4\n590=2*5*59:8\n591=3*197:4\n592=16*37=2^4*37:10\n593:2\n594=2*27*11=2*3^3*11:16\n595=5*7*17:8\n596=4*149:6\n597=3*199:4\n598=2*13*23:8\n599:2\n600=2^3*3*5^2:24\n\nContinue from s(550)=27554, M=7.\nn=551: d=4. s=27554+28=27582. M=8.\nn=552: d=16. s=27582+112=27694. M=9.\nn=553: d=4. s=27694+36=27730. M=7.\nn=554: d=4. s=27730+28=27758. M=8.\nn=555: d=8. s=27758+64=27822. M=8.\nn=556: d=6. s=27822+48=27870. M=8.\nn=557: d=2. s=27870+16=27886. M=8.\nn=558: d=12. s=27886+96=27982. M=9.\nn=559: d=4. s=27982+36=28018. M=8.\nn=560: d=20. s=28018+160=28178. M=8.\nn=561: d=8. s=28178+64=28242. M=8.\nn=562: d=4. s=28242+32=28274. M=8.\nn=563: d=2. s=28274+16=28290. M=9.\nn=564: d=12. s=28290+108=28398. M=9.\nn=565: d=4. s=28398+36=28434. M=8.\nn=566: d=4. s=28434+32=28466. M=8.\nn=567: d=10. s=28466+80=28546. M=8.\nn=568: d=8. s=28546+64=28610. M=8.\nn=569: d=2. s=28610+16=28626. M=8.\nn=570: d=16. s=28626+128=28754. M=8.\nn=571: d=2. s=28754+16=28770. M=8.\nn=572: d=12. s=28770+96=28866. M=8.\nn=573: d=4. s=28866+32=28898. M=9.\nn=574: d=8. s=28898+72=28970. M=9.\nn=575: d=6. s=28970+54=29024. M=9.\nn=576: d=21. s=29024+189=29213. M=9.\nn=577: d=2. s=29213+18=29231. M=9.\nn=578: d=6. s=29231+54=29285. M=9.\nn=579: d=4. s=29285+36=29321. M=9.\nn=580: d=12. s=29321+108=29429. M=9.\nn=581: d=4. s=29429+36=29465. M=9.\nn=582: d=8. s=29465+72=29537. M=9.\nn=583: d=4. s=29537+36=29573. M=9.\nn=584: d=8. s=29573+72=29645. M=9.\nn=585: d=12. s=29645+108=29753. M=9.\nn=586: d=4. s=29753+36=29789. M=9.\nn=587: d=2. s=29789+18=29807. M=9.\nn=588: d=18. s=29807+162=29969. M=9.\nn=589: d=4. s=29969+36=30005. M=5.\nn=590: d=8. s=30005+40=30045. M=5.\nn=591: d=4. s=30045+20=30065. M=6.\nn=592: d=10. s=30065+60=30125. M=5.\nn=593: d=2. s=30125+10=30135. M=5.\nn=594: d=16. s=30135+80=30215. M=5.\nn=595: d=8. s=30215+40=30255. M=5.\nn=596: d=6. s=30255+30=30285. M=8.\nn=597: d=4. s=30285+32=30317. M=7.\nn=598: d=8. s=30317+56=30373. M=7.\nn=599: d=2. s=30373+14=30387. M=8.\nn=600: d=24. s=30387+192=30579. M=9.\n\ns(600)=30579, M=9.\n\nGrowth: from 27554 to 30579 over 50 steps = 3025, ~60/step now. \n\nContinue. d(n) 601..650:\n601:2\n602=2*7*43:8\n603=9*67=3^2*67:6\n604=4*151:6\n605=5*121=5*11^2:6\n606=2*3*101:8\n607:2\n608=2^5*19:12\n609=3*7*29:8\n610=2*5*61:8\n611=13*47:4\n612=4*9*17=2^2*3^2*17:18\n613:2\n614=2*307:4\n615=3*5*41:8\n616=8*7*11=2^3*7*11:16\n617:2\n618=2*3*103:8\n619:2\n620=4*5*31=2^2*5*31:12\n621=27*23=3^3*23:8\n622=2*311:4\n623=7*89:4\n624=2^4*3*13:20\n625=5^4:5\n626=2*313:4\n627=3*11*19:8\n628=4*157:6\n629=17*37:4\n630=2*3^2*5*7:24\n631:2\n632=8*79:8\n633=3*211:4\n634=2*317:4\n635=5*127:4\n636=4*3*53=2^2*3*53:12\n637=7^2*13:6\n638=2*11*29:8\n639=9*71=3^2*71:6\n640=2^7*5:16\n641:2\n642=2*3*107:8\n643:2\n644=4*7*23=2^2*7*23:12\n645=3*5*43:8\n646=2*17*19:8\n647:2\n648=2^3*3^4:20\n649=11*59:4\n650=2*25*13=2*5^2*13:12\n\nContinue from s(600)=30579, M=9.\nn=601: d=2. s=30579+18=30597. M=9.\nn=602: d=8. s=30597+72=30669. M=9.\nn=603: d=6. s=30669+54=30723. M=7.\nn=604: d=6. s=30723+42=30765. M=7.\nn=605: d=6. s=30765+42=30807. M=8.\nn=606: d=8. s=30807+64=30871. M=8.\nn=607: d=2. s=30871+16=30887. M=8.\nn=608: d=12. s=30887+96=30983. M=9.\nn=609: d=8. s=30983+72=31055. M=5.\nn=610: d=8. s=31055+40=31095. M=9.\nn=611: d=4. s=31095+36=31131. M=3.\nn=612: d=18. s=31131+54=31185. M=8.\nn=613: d=2. s=31185+16=31201. M=3.\nn=614: d=4. s=31201+12=31213. M=3.\nn=615: d=8. s=31213+24=31237. M=7.\nn=616: d=16. s=31237+112=31349. M=9.\nn=617: d=2. s=31349+18=31367. M=7.\nn=618: d=8. s=31367+56=31423. M=4.\nn=619: d=2. s=31423+8=31431. M=4.\nn=620: d=12. s=31431+48=31479. M=9.\nn=621: d=8. s=31479+72=31551. M=5.\nn=622: d=4. s=31551+20=31571. M=7.\nn=623: d=4. s=31571+28=31599. M=9.\nn=624: d=20. s=31599+180=31779. M=9.\nn=625: d=5. s=31779+45=31824. M=8.\nn=626: d=4. s=31824+32=31856. M=8.\nn=627: d=8. s=31856+64=31920. M=9.\nn=628: d=6. s=31920+54=31974. M=9.\nn=629: d=4. s=31974+36=32010. M=3.\nn=630: d=24. s=32010+72=32082. M=8.\nn=631: d=2. s=32082+16=32098. M=9.\nn=632: d=8. s=32098+72=32170. M=7.\nn=633: d=4. s=32170+28=32198. M=9.\nn=634: d=4. s=32198+36=32234. M=4.\nn=635: d=4. s=32234+16=32250. M=5.\nn=636: d=12. s=32250+60=32310. M=3.\nn=637: d=6. s=32310+18=32328. M=8.\nn=638: d=8. s=32328+64=32392. M=9.\nn=639: d=6. s=32392+54=32446. M=6.\nn=640: d=16. s=32446+96=32542. M=5.\nn=641: d=2. s=32542+10=32552. M=5.\nn=642: d=8. s=32552+40=32592. M=9.\nn=643: d=2. s=32592+18=32610. M=6.\nn=644: d=12. s=32610+72=32682. M=8.\nn=645: d=8. s=32682+64=32746. M=7.\nn=646: d=8. s=32746+56=32802. M=8.\nn=647: d=2. s=32802+16=32818. M=8.\nn=648: d=20. s=32818+160=32978. M=9.\nn=649: d=4. s=32978+36=33014. M=4.\nn=650: d=12. s=33014+48=33062. M=6.\n\ns(650)=33062, M=6.\n\nContinue. d(n) 651..700:\n651=3*7*31:8\n652=4*163:6\n653:2\n654=2*3*109:8\n655=5*131:4\n656=16*41=2^4*41:10\n657=9*73=3^2*73:6\n658=2*7*47:8\n659:2\n660=2^2*3*5*11:24\n661:2\n662=2*331:4\n663=3*13*17:8\n664=8*83:8\n665=5*7*19:8\n666=2*9*37=2*3^2*37:12\n667=23*29:4\n668=4*167:6\n669=3*223:4\n670=2*5*67:8\n671=11*61:4\n672=2^5*3*7:24\n673:2\n674=2*337:4\n675=27*25=3^3*5^2:12\n676=4*169=2^2*13^2:9\n677:2\n678=2*3*113:8\n679=7*97:4\n680=8*5*17=2^3*5*17:16\n681=3*227:4\n682=2*11*31:8\n683:2\n684=4*9*19=2^2*3^2*19:18\n685=5*137:4\n686=2*343=2*7^3:8\n687=3*229:4\n688=16*43=2^4*43:10\n689=13*53:4\n690=2*3*5*23:16\n691:2\n692=4*173:6\n693=9*7*11=3^2*7*11:12\n694=2*347:4\n695=5*139:4\n696=8*3*29=2^3*3*29:16\n697=17*41:4\n698=2*349:4\n699=3*233:4\n700=2^2*5^2*7:18\n\nContinue from s(650)=33062, M=6.\nn=651: d=8. s=33062+48=33110. M=3.\nn=652: d=6. s=33110+18=33128. M=8.\nn=653: d=2. s=33128+16=33144. M=4.\nn=654: d=8. s=33144+32=33176. M=7.\nn=655: d=4. s=33176+28=33204. M=4.\nn=656: d=10. s=33204+40=33244. M=4.\nn=657: d=6. s=33244+24=33268. M=8.\nn=658: d=8. s=33268+64=33332. M=3.\nn=659: d=2. s=33332+6=33338. M=8.\nn=660: d=24. s=33338+192=33530. M=5.\nn=661: d=2. s=33530+10=33540. M=5.\nn=662: d=4. s=33540+20=33560. M=6.\nn=663: d=8. s=33560+48=33608. M=8.\nn=664: d=8. s=33608+64=33672. M=7.\nn=665: d=8. s=33672+56=33728. M=8.\nn=666: d=12. s=33728+96=33824. M=8.\nn=667: d=4. s=33824+32=33856. M=8.\nn=668: d=6. s=33856+48=33904. M=9.\nn=669: d=4. s=33904+36=33940. M=9.\nn=670: d=8. s=33940+72=34012. M=4.\nn=671: d=4. s=34012+16=34028. M=8.\nn=672: d=24. s=34028+192=34220. M=4.\nn=673: d=2. s=34220+8=34228. M=8.\nn=674: d=4. s=34228+32=34260. M=6.\nn=675: d=12. s=34260+72=34332. M=4.\nn=676: d=9. s=34332+36=34368. M=8.\nn=677: d=2. s=34368+16=34384. M=8.\nn=678: d=8. s=34384+64=34448. M=8.\nn=679: d=4. s=34448+32=34480. M=8.\nn=680: d=16. s=34480+128=34608. M=8.\nn=681: d=4. s=34608+32=34640. M=6.\nn=682: d=8. s=34640+48=34688. M=8.\nn=683: d=2. s=34688+16=34704. M=7.\nn=684: d=18. s=34704+126=34830. M=8.\nn=685: d=4. s=34830+32=34862. M=8.\nn=686: d=8. s=34862+64=34926. M=9.\nn=687: d=4. s=34926+36=34962. M=9.\nn=688: d=10. s=34962+90=35052. M=5.\nn=689: d=4. s=35052+20=35072. M=7.\nn=690: d=16. s=35072+112=35184. M=8.\nn=691: d=2. s=35184+16=35200. M=5.\nn=692: d=6. s=35200+30=35230. M=5.\nn=693: d=12. s=35230+60=35290. M=9.\nn=694: d=4. s=35290+36=35326. M=6.\nn=695: d=4. s=35326+24=35350. M=5.\nn=696: d=16. s=35350+80=35430. M=5.\nn=697: d=4. s=35430+20=35450. M=5.\nn=698: d=4. s=35450+20=35470. M=7.\nn=699: d=4. s=35470+28=35498. M=9.\nn=700: d=18. s=35498+162=35660. M=6.\n\ns(700)=35660, M=6.\n\nContinue. d(n) 701..750:\n701:2\n702=2*27*13=2*3^3*13:16\n703=19*37:4\n704=2^6*11:14\n705=3*5*47:8\n706=2*353:4\n707=7*101:4\n708=4*3*59=2^2*3*59:12\n709:2\n710=2*5*71:8\n711=9*79=3^2*79:6\n712=8*89:8\n713=23*31:4\n714=2*3*7*17:16\n715=5*11*13:8\n716=4*179:6\n717=3*239:4\n718=2*359:4\n719:2\n720=2^4*3^2*5:30\n721=7*103:4\n722=2*361=2*19^2:6\n723=3*241:4\n724=4*181:6\n725=25*29=5^2*29:6\n726=2*3*121=2*3*11^2:12\n727:2\n728=8*7*13=2^3*7*13:16\n729=3^6:7\n730=2*5*73:8\n731=17*43:4\n732=4*3*61=2^2*3*61:12\n733:2\n734=2*367:4\n735=3*5*49=3*5*7^2:12\n736=2^5*23:12\n737=11*67:4\n738=2*9*41=2*3^2*41:12\n739:2\n740=4*5*37=2^2*5*37:12\n741=3*13*19:8\n742=2*7*53:8\n743:2\n744=8*3*31=2^3*3*31:16\n745=5*149:4\n746=2*373:4\n747=9*83=3^2*83:6\n748=4*11*17=2^2*11*17:12\n749=7*107:4\n750=2*3*125=2*3*5^3:16\n\nContinue from s(700)=35660, M=6.\nn=701: d=2. s=35660+12=35672. M=7.\nn=702: d=16. s=35672+112=35784. M=8.\nn=703: d=4. s=35784+32=35816. M=8.\nn=704: d=14. s=35816+112=35928. M=9.\nn=705: d=8. s=35928+72=36000. M=6.\nn=706: d=4. s=36000+24=36024. M=6.\nn=707: d=4. s=36024+24=36048. M=8.\nn=708: d=12. s=36048+96=36144. M=6.\nn=709: d=2. s=36144+12=36156. M=6.\nn=710: d=8. s=36156+48=36204. M=6.\nn=711: d=6. s=36204+36=36240. M=6.\nn=712: d=8. s=36240+48=36288. M=8.\nn=713: d=4. s=36288+32=36320. M=6.\nn=714: d=16. s=36320+96=36416. M=6.\nn=715: d=8. s=36416+48=36464. M=6.\nn=716: d=6. s=36464+36=36500. M=6.\nn=717: d=4. s=36500+24=36524. M=6.\nn=718: d=4. s=36524+24=36548. M=8.\nn=719: d=2. s=36548+16=36564. M=6.\nn=720: d=30. s=36564+180=36744. M=7.\nn=721: d=4. s=36744+28=36772. M=7.\nn=722: d=6. s=36772+42=36814. M=8.\nn=723: d=4. s=36814+32=36846. M=8.\nn=724: d=6. s=36846+48=36894. M=9.\nn=725: d=6. s=36894+54=36948. M=9.\nn=726: d=12. s=36948+108=37056. M=7.\nn=727: d=2. s=37056+14=37070. M=7.\nn=728: d=16. s=37070+112=37182. M=8.\nn=729: d=7. s=37182+56=37238. M=8.\nn=730: d=8. s=37238+64=37302. M=7.\nn=731: d=4. s=37302+28=37330. M=7.\nn=732: d=12. s=37330+84=37414. M=7.\nn=733: d=2. s=37414+14=37428. M=8.\nn=734: d=4. s=37428+32=37460. M=7.\nn=735: d=12. s=37460+84=37544. M=7.\nn=736: d=12. s=37544+84=37628. M=8.\nn=737: d=4. s=37628+32=37660. M=7.\nn=738: d=12. s=37660+84=37744. M=7.\nn=739: d=2. s=37744+14=37758. M=8.\nn=740: d=12. s=37758+96=37854. M=8.\nn=741: d=8. s=37854+64=37918. M=9.\nn=742: d=8. s=37918+72=37990. M=9.\nn=743: d=2. s=37990+18=38008. M=8.\nn=744: d=16. s=38008+128=38136. M=8.\nn=745: d=4. s=38136+32=38168. M=8.\nn=746: d=4. s=38168+32=38200. M=8.\nn=747: d=6. s=38200+48=38248. M=8.\nn=748: d=12. s=38248+96=38344. M=8.\nn=749: d=4. s=38344+32=38376. M=8.\nn=750: d=16. s=38376+128=38504. M=8.\n\ns(750)=38504, M=8.\n\nContinue. d(n) 751..800:\n751:2\n752=16*47=2^4*47:10\n753=3*251:4\n754=2*13*29:8\n755=5*151:4\n756=4*27*7=2^2*3^3*7:24\n757:2\n758=2*379:4\n759=3*11*23:8\n760=8*5*19=2^3*5*19:16\n761:2\n762=2*3*127:8\n763=7*109:4\n764=4*191:6\n765=9*5*17=3^2*5*17:12\n766=2*383:4\n767=13*59:4\n768=2^8*3:18\n769:2\n770=2*5*7*11:16\n771=3*257:4\n772=4*193:6\n773:2\n774=2*9*43=2*3^2*43:12\n775=25*31=5^2*31:6\n776=8*97:8\n777=3*7*37:8\n778=2*389:4\n779=19*41:4\n780=2^2*3*5*13:24\n781=11*71:4\n782=2*17*23:8\n783=27*29=3^3*29:8\n784=16*49=2^4*7^2:15\n785=5*157:4\n786=2*3*131:8\n787:2\n788=4*197:6\n789=3*263:4\n790=2*5*79:8\n791=7*113:4\n792=8*9*11=2^3*3^2*11:24\n793=13*61:4\n794=2*397:4\n795=3*5*53:8\n796=4*199:6\n797:2\n798=2*3*7*19:16\n799=17*47:4\n800=2^5*5^2:18\n\nContinue from s(750)=38504, M=8.\nn=751: d=2. s=38504+16=38520. M=8.\nn=752: d=10. s=38520+80=38600. M=8.\nn=753: d=4. s=38600+32=38632. M=8.\nn=754: d=8. s=38632+64=38696. M=9.\nn=755: d=4. s=38696+36=38732. M=8.\nn=756: d=24. s=38732+192=38924. M=9.\nn=757: d=2. s=38924+18=38942. M=9.\nn=758: d=4. s=38942+36=38978. M=9.\nn=759: d=8. s=38978+72=39050. M=9.\nn=760: d=16. s=39050+144=39194. M=9.\nn=761: d=2. s=39194+18=39212. M=9.\nn=762: d=8. s=39212+72=39284. M=9.\nn=763: d=4. s=39284+36=39320. M=9.\nn=764: d=6. s=39320+54=39374. M=9.\nn=765: d=12. s=39374+108=39482. M=9.\nn=766: d=4. s=39482+36=39518. M=9.\nn=767: d=4. s=39518+36=39554. M=9.\nn=768: d=18. s=39554+162=39716. M=9.\nn=769: d=2. s=39716+18=39734. M=9.\nn=770: d=16. s=39734+144=39878. M=9.\nn=771: d=4. s=39878+36=39914. M=9.\nn=772: d=6. s=39914+54=39968. M=9.\nn=773: d=2. s=39968+18=39986. M=9.\nn=774: d=12. s=39986+108=40094. M=9.\nn=775: d=6. s=40094+54=40148. M=8.\nn=776: d=8. s=40148+64=40212. M=4.\nn=777: d=8. s=40212+32=40244. M=4.\nn=778: d=4. s=40244+16=40260. M=6.\nn=779: d=4. s=40260+24=40284. M=8.\nn=780: d=24. s=40284+192=40476. M=7.\nn=781: d=4. s=40476+28=40504. M=5.\nn=782: d=8. s=40504+40=40544. M=5.\nn=783: d=8. s=40544+40=40584. M=8.\nn=784: d=15. s=40584+120=40704. M=7.\nn=785: d=4. s=40704+28=40732. M=7.\nn=786: d=8. s=40732+56=40788. M=8.\nn=787: d=2. s=40788+16=40804. M=8.\nn=788: d=6. s=40804+48=40852. M=8.\nn=789: d=4. s=40852+32=40884. M=8.\nn=790: d=8. s=40884+64=40948. M=9.\nn=791: d=4. s=40948+36=40984. M=9.\nn=792: d=24. s=40984+216=41200. M=4.\nn=793: d=4. s=41200+16=41216. M=6.\nn=794: d=4. s=41216+24=41240. M=4.\nn=795: d=8. s=41240+32=41272. M=7.\nn=796: d=6. s=41272+42=41314. M=4.\nn=797: d=2. s=41314+8=41322. M=4.\nn=798: d=16. s=41322+64=41386. M=8.\nn=799: d=4. s=41386+32=41418. M=8.\nn=800: d=18. s=41418+144=41562. M=6.\n\ns(800)=41562, M=6.\n\nContinue. d(n) 801..850:\n801=9*89=3^2*89:6\n802=2*401:4\n803=11*73:4\n804=4*3*67=2^2*3*67:12\n805=5*7*23:8\n806=2*13*31:8\n807=3*269:4\n808=8*101:8\n809:2\n810=2*3^4*5:20\n811:2\n812=4*7*29=2^2*7*29:12\n813=3*271:4\n814=2*11*37:8\n815=5*163:4\n816=2^4*3*17:20\n817=19*43:4\n818=2*409:4\n819=9*7*13=3^2*7*13:12\n820=4*5*41=2^2*5*41:12\n821:2\n822=2*3*137:8\n823:2\n824=8*103:8\n825=3*25*11=3*5^2*11:12\n826=2*7*59:8\n827:2\n828=4*9*23=2^2*3^2*23:18\n829:2\n830=2*5*83:8\n831=3*277:4\n832=2^6*13:14\n833=7^2*17:6\n834=2*3*139:8\n835=5*167:4\n836=4*11*19=2^2*11*19:12\n837=27*31=3^3*31:8\n838=2*419:4\n839:2\n840=2^3*3*5*7:32\n841=29^2:3\n842=2*421:4\n843=3*281:4\n844=4*211:6\n845=5*169=5*13^2:6\n846=2*9*47=2*3^2*47:12\n847=7*121=7*11^2:6\n848=16*53=2^4*53:10\n849=3*283:4\n850=2*25*17=2*5^2*17:12\n\nContinue from s(800)=41562, M=6.\nn=801: d=6. s=41562+36=41598. M=9.\nn=802: d=4. s=41598+36=41634. M=6.\nn=803: d=4. s=41634+24=41658. M=8.\nn=804: d=12. s=41658+96=41754. M=7.\nn=805: d=8. s=41754+56=41810. M=8.\nn=806: d=8. s=41810+64=41874. M=8.\nn=807: d=4. s=41874+32=41906. M=9.\nn=808: d=8. s=41906+72=41978. M=9.\nn=809: d=2. s=41978+18=41996. M=9.\nn=810: d=20. s=41996+180=42176. M=7.\nn=811: d=2. s=42176+14=42190. M=9.\nn=812: d=12. s=42190+108=42298. M=9.\nn=813: d=4. s=42298+36=42334. M=4.\nn=814: d=8. s=42334+32=42366. M=6.\nn=815: d=4. s=42366+24=42390. M=9.\nn=816: d=20. s=42390+180=42570. M=7.\nn=817: d=4. s=42570+28=42598. M=9.\nn=818: d=4. s=42598+36=42634. M=6.\nn=819: d=12. s=42634+72=42706. M=7.\nn=820: d=12. s=42706+84=42790. M=9.\nn=821: d=2. s=42790+18=42808. M=8.\nn=822: d=8. s=42808+64=42872. M=8.\nn=823: d=2. s=42872+16=42888. M=8.\nn=824: d=8. s=42888+64=42952. M=9.\nn=825: d=12. s=42952+108=43060. M=6.\nn=826: d=8. s=43060+48=43108. M=8.\nn=827: d=2. s=43108+16=43124. M=4.\nn=828: d=18. s=43124+72=43196. M=9.\nn=829: d=2. s=43196+18=43214. M=4.\nn=830: d=8. s=43214+32=43246. M=6.\nn=831: d=4. s=43246+24=43270. M=7.\nn=832: d=14. s=43270+98=43368. M=8.\nn=833: d=6. s=43368+48=43416. M=6.\nn=834: d=8. s=43416+48=43464. M=6.\nn=835: d=4. s=43464+24=43488. M=8.\nn=836: d=12. s=43488+96=43584. M=8.\nn=837: d=8. s=43584+64=43648. M=8.\nn=838: d=4. s=43648+32=43680. M=8.\nn=839: d=2. s=43680+16=43696. M=9.\nn=840: d=32. s=43696+288=43984. M=9.\nn=841: d=3. s=43984+27=44011. M=4.\nn=842: d=4. s=44011+16=44027. M=7.\nn=843: d=4. s=44027+28=44055. M=5.\nn=844: d=6. s=44055+30=44085. M=8.\nn=845: d=6. s=44085+48=44133. M=4.\nn=846: d=12. s=44133+48=44181. M=8.\nn=847: d=6. s=44181+48=44229. M=9.\nn=848: d=10. s=44229+90=44319. M=9.\nn=849: d=4. s=44319+36=44355. M=5.\nn=850: d=12. s=44355+60=44415. M=5.\n\ns(850)=44415, M=5.\n\nContinue. d(n) 851..900:\n851=23*37:4\n852=4*3*71=2^2*3*71:12\n853:2\n854=2*7*61:8\n855=9*5*19=3^2*5*19:12\n856=8*107:8\n857:2\n858=2*3*11*13:16\n859:2\n860=4*5*43=2^2*5*43:12\n861=3*7*41:8\n862=2*431:4\n863:2\n864=2^5*27=2^5*3^3:24\n865=5*173:4\n866=2*433:4\n867=3*289=3*17^2:6\n868=4*7*31=2^2*7*31:12\n869=11*79:4\n870=2*3*5*29:16\n871=13*67:4\n872=8*109:8\n873=9*97=3^2*97:6\n874=2*19*23:8\n875=125*7=5^3*7:8\n876=4*3*73=2^2*3*73:12\n877:2\n878=2*439:4\n879=3*293:4\n880=2^4*5*11:20\n881:2\n882=2*9*49=2*3^2*7^2:18\n883:2\n884=4*13*17=2^2*13*17:12\n885=3*5*59:8\n886=2*443:4\n887:2\n888=8*3*37=2^3*3*37:16\n889=7*127:4\n890=2*5*89:8\n891=81*11=3^4*11:10\n892=4*223:6\n893=19*47:4\n894=2*3*149:8\n895=5*179:4\n896=2^7*7:16\n897=3*13*23:8\n898=2*449:4\n899=29*31:4\n900=2^2*3^2*5^2:27\n\nContinue from s(850)=44415, M=5.\nn=851: d=4. s=44415+20=44435. M=5.\nn=852: d=12. s=44435+60=44495. M=9.\nn=853: d=2. s=44495+18=44513. M=5.\nn=854: d=8. s=44513+40=44553. M=5.\nn=855: d=12. s=44553+60=44613. M=6.\nn=856: d=8. s=44613+48=44661. M=6.\nn=857: d=2. s=44661+12=44673. M=7.\nn=858: d=16. s=44673+112=44785. M=8.\nn=859: d=2. s=44785+16=44801. M=8.\nn=860: d=12. s=44801+96=44897. M=9.\nn=861: d=8. s=44897+72=44969. M=9.\nn=862: d=4. s=44969+36=45005. M=5.\nn=863: d=2. s=45005+10=45015. M=5.\nn=864: d=24. s=45015+120=45135. M=5.\nn=865: d=4. s=45135+20=45155. M=5.\nn=866: d=4. s=45155+20=45175. M=7.\nn=867: d=6. s=45175+42=45217. M=7.\nn=868: d=12. s=45217+84=45301. M=5.\nn=869: d=4. s=45301+20=45321. M=5.\nn=870: d=16. s=45321+80=45401. M=5.\nn=871: d=4. s=45401+20=45421. M=5.\nn=872: d=8. s=45421+40=45461. M=6.\nn=873: d=6. s=45461+36=45497. M=9.\nn=874: d=8. s=45497+72=45569. M=9.\nn=875: d=8. s=45569+72=45641. M=6.\nn=876: d=12. s=45641+72=45713. M=7.\nn=877: d=2. s=45713+14=45727. M=7.\nn=878: d=4. s=45727+28=45755. M=7.\nn=879: d=4. s=45755+28=45783. M=8.\nn=880: d=20. s=45783+160=45943. M=9.\nn=881: d=2. s=45943+18=45961. M=9.\nn=882: d=18. s=45961+162=46123. M=6.\nn=883: d=2. s=46123+12=46135. M=6.\nn=884: d=12. s=46135+72=46207. M=7.\nn=885: d=8. s=46207+56=46263. M=6.\nn=886: d=4. s=46263+24=46287. M=8.\nn=887: d=2. s=46287+16=46303. M=6.\nn=888: d=16. s=46303+96=46399. M=9.\nn=889: d=4. s=46399+36=46435. M=6.\nn=890: d=8. s=46435+48=46483. M=8.\nn=891: d=10. s=46483+80=46563. M=6.\nn=892: d=6. s=46563+36=46599. M=9.\nn=893: d=4. s=46599+36=46635. M=6.\nn=894: d=8. s=46635+48=46683. M=8.\nn=895: d=4. s=46683+32=46715. M=7.\nn=896: d=16. s=46715+112=46827. M=8.\nn=897: d=8. s=46827+64=46891. M=9.\nn=898: d=4. s=46891+36=46927. M=9.\nn=899: d=4. s=46927+36=46963. M=9.\nn=900: d=27. s=46963+243=47206. M=7.\n\ns(900)=47206, M=7.\n\nContinue. d(n) 901..950:\n901=17*53:4\n902=2*11*41:8\n903=3*7*43:8\n904=8*113:8\n905=5*181:4\n906=2*3*151:8\n907:2\n908=4*227:6\n909=9*101=3^2*101:6\n910=2*5*7*13:16\n911:2\n912=2^4*3*19:20\n913=11*83:4\n914=2*457:4\n915=3*5*61:8\n916=4*229:6\n917=7*131:4\n918=2*27*17=2*3^3*17:16\n919:2\n920=8*5*23=2^3*5*23:16\n921=3*307:4\n922=2*461:4\n923=13*71:4\n924=2^2*3*7*11:24\n925=25*37=5^2*37:6\n926=2*463:4\n927=9*103=3^2*103:6\n928=2^5*29:12\n929:2\n930=2*3*5*31:16\n931=7^2*19:6\n932=4*233:6\n933=3*311:4\n934=2*467:4\n935=5*11*17:8\n936=2^3*3^2*13:24\n937:2\n938=2*7*67:8\n939=3*313:4\n940=4*5*47=2^2*5*47:12\n941:2\n942=2*3*157:8\n943=23*41:4\n944=16*59=2^4*59:10\n945=27*5*7=3^3*5*7:16\n946=2*11*43:8\n947:2\n948=4*3*79=2^2*3*79:12\n949=13*73:4\n950=2*25*19=2*5^2*19:12\n\nContinue from s(900)=47206, M"
    },
    {
      "task_id": "PILOT-Q1",
      "task_type": "simple_qa",
      "answer": "10152",
      "gold": "10152",
      "correct": true,
      "tokens_in": 222,
      "tokens_out": 211,
      "cost_usd": 0.006385,
      "latency_s": 3.34,
      "timed_out": false,
      "solution": "Monday: 48\nTuesday: 54\nWednesday: 60\nThursday: 66\nFriday: 72\n\nMonday: 48\u00d736 = 1728\nTuesday: 54\u00d736 = 1944\nWednesday: 60 crates, 1/3 = 20 half-full (18 each), 40 full (36 each)\n= 20\u00d718 + 40\u00d736 = 360 + 1440 = 1800\nThursday: 66\u00d736 = 2376\nFriday: 72 crates, 8 returned \u2192 64\u00d736 = 2304\n\nTotal: 1728+1944+1800+2376+2304 = 10152\n\nANSWER: 10152"
    },
    {
      "task_id": "PILOT-R1",
      "task_type": "reasoning",
      "answer": "12",
      "gold": "12",
      "correct": true,
      "tokens_in": 260,
      "tokens_out": 2919,
      "cost_usd": 0.074275,
      "latency_s": 30.52,
      "timed_out": false,
      "solution": "<think>\nLet me denote positions 1-5. Two liars finish in consecutive positions. Ava ahead of Dan (Ava position < Dan position).\n\nStatements:\n- Ava: \"I finished immediately ahead of Ben\" (Ava = Ben - 1)\n- Ben: \"Cleo finished last\" (Cleo = 5)\n- Cleo: \"I finished ahead of Ava\" (Cleo < Ava)\n- Dan: \"Eve did not finish first\" (Eve \u2260 1)\n- Eve: \"Dan finished third\" (Dan = 3)\n\nExactly 2 liars, they finish consecutive positions.\n\nLet me try to figure out consistency.\n\nNote Cleo says Cleo < Ava. Ben says Cleo = 5. If both Ben and Cleo tell truth, contradiction (Cleo=5 can't be ahead of Ava). So at least one of Ben, Cleo lies.\n\nLet me consider cases based on who the two liars are.\n\nLet me think about who tells truth. Let's try each pair of liars being consecutive in finishing position.\n\nActually let me just try to find a full assignment.\n\nLet me enumerate. Constraints from truths, negations from lies.\n\nLet me consider the liars set L (size 2), consecutive positions.\n\nLet me try assuming truth-tellers and check.\n\nSince exactly one of Ben/Cleo lies at least, let's consider subcases.\n\nCase A: Ben lies, Cleo tells truth.\nThen Cleo < Ava is true. Ben's statement Cleo=5 is false, so Cleo\u22605 (consistent with Cleo<Ava).\nThe other liar is among Ava, Dan, Eve.\n\nCase B: Cleo lies, Ben tells truth.\nThen Cleo=5 true, Cleo<Ava false so Ava<Cleo=5 fine. Cleo=5.\nOther liar among Ava, Dan, Eve.\n\nCase C: both Ben and Cleo lie.\nThen they're the two liars, must be consecutive positions. Ben's false: Cleo\u22605. Cleo's false: Ava<Cleo (Ava ahead of Cleo). Ava, Dan, Eve tell truth.\nAva true: Ava=Ben-1. Dan true: Eve\u22601. Eve true: Dan=3.\nSo Dan=3, Ava<Dan means Ava\u2208{1,2}. Ava=Ben-1.\nAva<Cleo. Eve\u22601.\nBen and Cleo consecutive positions.\nPositions: Dan=3. Remaining 1,2,4,5 for Ava,Ben,Cleo,Eve.\nAva=Ben-1. Possible: Ava=1,Ben=2; Ava=4,Ben=5 (but Ava<3), so Ava=1,Ben=2.\nThen Cleo,Eve in {4,5}. Ava<Cleo yes. Eve\u22601 yes. Cleo\u22605 required, so Cleo=4, Eve=5.\nBen and Cleo positions: Ben=2, Cleo=4, not consecutive. Contradiction (liars must be consecutive).\nSo Case C fails.\n\nNow Case A: Ben lies, Cleo truth. Second liar among Ava,Dan,Eve.\nCleo<Ava true. Ava<Dan.\nTruth-tellers among the others tell truth.\n\nSub A1: liars = Ben and Ava.\nThen Ava lies: Ava\u2260Ben-1. Ben lies: Cleo\u22605.\nTruth: Cleo (Cleo<Ava), Dan (Eve\u22601), Eve (Dan=3).\nDan=3, Ava<3 so Ava\u2208{1,2}. Cleo<Ava so Cleo< Ava, meaning Cleo=1,Ava=2. Ava=2,Cleo=1.\nRemaining positions 4,5 for Ben,Eve. Eve\u22601 ok. Cleo\u22605 ok.\nAva\u2260Ben-1: Ben is 4 or5, Ava=2, Ben-1=3 or4, \u22602 fine.\nLiars Ben & Ava must be consecutive: Ava=2, Ben\u2208{4,5}. Not consecutive. Fail.\n\nSub A2: liars = Ben and Dan.\nDan lies: Eve=1 (since \"Eve did not finish first\" false \u2192 Eve=1).\nBen lies: Cleo\u22605.\nTruth: Ava (Ava=Ben-1), Cleo (Cleo<Ava), Eve (Dan=3).\nEve=1, Dan=3. Ava<Dan ok.\nAva=Ben-1. Cleo<Ava.\nPositions: Eve=1, Dan=3. Remaining 2,4,5 for Ava,Ben,Cleo.\nAva=Ben-1: possible Ava=4,Ben=5. Then Cleo=2. Cleo<Ava (2<4) ok. Cleo\u22605 ok.\nCheck Ava<Dan? Ava=4, Dan=3, 4<3 false! Violates Ava ahead of Dan. Fail.\nOther: Ava,Ben consecutive from {2,4,5}: Ava=4,Ben=5 only (since 2&3 no, need both in set... Ava=2,Ben=3 but 3 is Dan). So fail.\n\nSub A3: liars = Ben and Eve.\nEve lies: Dan\u22603. Ben lies: Cleo\u22605.\nTruth: Ava (Ava=Ben-1), Cleo (Cleo<Ava), Dan (Eve\u22601).\nAva<Dan, Cleo<Ava, so Cleo<Ava<Dan.\nAva=Ben-1.\nBen,Eve liars consecutive.\nLet's find positions. Cleo<Ava<Dan. Ava=Ben-1.\nLet me enumerate positions.\nVariables A,B,C,D,E permutation of 1-5.\nConstraints: C<A<D, A=B-1, Dan\u22603, Cleo\u22605, Eve\u22601, Ben&Eve consecutive positions (|B-E|=1).\nA=B-1 so B=A+1.\nC<A<D.\nLet me try A values.\nA must be >C\u22651, so A\u22652. A<D.\nIf A=2: B=3. C<2 so C=1. D>2, D\u2208{4,5}, E remaining.\n Positions used A=2,B=3,C=1. D,E from {4,5}. Dan\u22603 ok. Cleo=1\u22605 ok. Eve\u22601 ok (E is 4or5).\n Ben&Eve consecutive: B=3, E\u2208{4,5}. E=4 consecutive to 3. Then D=5.\n Check: A=2,B=3,C=1,D=5,E=4. C<A<D: 1<2<5 ok. Dan=5\u22603 ok. \n Liars Ben(3),Eve(4) consecutive \u2713.\n All good! Let me verify all statements.\n Ava(2): \"immediately ahead of Ben\" Ava=2, Ben=3, yes Ava=Ben-1. Ava tells truth \u2713 (Ava is truth-teller).\n Ben(3): \"Cleo last\" Cleo=1, false. Ben liar \u2713.\n Cleo(1): \"ahead of Ava\" 1<2 true. Cleo truth \u2713.\n Dan(5): \"Eve not first\" Eve=4, true. Dan truth \u2713.\n Eve(4): \"Dan third\" Dan=5, false. Eve liar \u2713.\n Liars: Ben, Eve at positions 3 and 4. Consecutive \u2713. Ava(2) ahead of Dan(5) \u2713.\n\nThis works! Product of liar positions = 3\u00d74 = 12.\n\nLet me just double check uniqueness isn't needed; we found valid. Product = 12.\n</think>\n\nThe positions are: Cleo=1, Ava=2, Ben=3, Eve=4, Dan=5.\n\n- Ava (truth): 2 immediately ahead of Ben (3) \u2713\n- Ben (liar): Cleo last? No (Cleo=1) \u2717\n- Cleo (truth): ahead of Ava (1<2) \u2713\n- Dan (truth): Eve not first (Eve=4) \u2713\n- Eve (liar): Dan third? No (Dan=5) \u2717\n\nThe liars are Ben (position 3) and Eve (position 4), which are consecutive, and Ava (2) is ahead of Dan (5).\n\nProduct = 3 \u00d7 4 = 12.\n\nANSWER: 12"
    },
    {
      "task_id": "PILOT-C1",
      "task_type": "code",
      "answer": "925",
      "gold": "300062",
      "correct": false,
      "tokens_in": 152,
      "tokens_out": 32768,
      "cost_usd": 0.81996,
      "latency_s": 327.24,
      "timed_out": false,
      "solution": "<think>\nWe need n, n+4, n+6 all prime.\n\nSo we need primes p, p+4, p+6.\n\nLet me think. n=7: 7,11,13 all prime. Yes.\nn=13: 13,17,19 all prime. Yes.\nn=37: 37,41,43 all prime. Yes.\nn=67: 67,71,73 all prime. Yes.\nn=97: 97,101,103 prime. Yes.\nn=103: 103,107,109 prime. Yes.\nn=193: 193,197,199 prime. Yes.\nn=223: 223,227,229 prime. Yes.\nn=277: 277,281,283 prime. Yes.\nn=307: 307,311,313 prime. Yes.\n\nThis is basically prime triplets of form (p, p+4, p+6). Let me recall these.\n\nPrime triplets (p, p+4, p+6): 7, 13, 37, 67, 97, 103, 193, 223, 277, 307, 457, 613, 823, 853, 877, 1087, 1297, 1423, 1483, ...\n\nThis is a lot to enumerate up to 300000. This is hard to do by hand exhaustively but let me try to be systematic. Actually this is a known problem; I need to count all.\n\nLet me consider modular constraints. n must be prime. Consider mod 3: n, n+4\u2261n+1, n+6\u2261n mod 3. So n and n+6 same mod 3. For both prime (and >3), n\u22620 mod3. n+4\u2261n+1. We need n, n+1, n... wait n+4 mod3 = n+1. So residues mod 3 of the three: n, n+1, n. For none divisible by 3 (assuming all >3), we need n\u22620 and n+1\u22620, i.e., n\u22620 and n\u22622 mod3, so n\u22611 mod3.\n\nSo n\u22611 mod 3.\n\nMod 5: n, n+4, n+6 \u2261 n, n+4, n+1 mod5. Need none \u22610 (for primes>5). So n\u22620, n\u22621, n\u22624 mod5. So n\u22612 or 3 mod5.\n\nHmm, this narrows but still many.\n\nGiven the difficulty, let me just recall/derive the sequence up to 300000. This is OEIS A022004 (initial members of prime triples (p, p+4, p+6)).\n\nLet me recall A022004 terms:\n7, 13, 37, 67, 97, 103, 193, 223, 277, 307, 457, 613, 823, 853, 877, 1087, 1297, 1423, 1483, 1663, 1693, 1783, 1867, 1993, 2083, 2137, 2377, 2683, 2707, 2797, 3163, 3253, 3457, 3463, 3847, 4153, 4513, 4783, 5227, 5413, 5437, 5647, 5653, 5827, 6547, 6553, 6653, 6823, 7057, 7873, 8287, 8353, 8677, 9433, 9643, 9007? ...\n\nThis is getting complicated and error-prone. This is truly a computational problem. I cannot reliably enumerate all by hand.\n\nLet me think differently - the answer asks A+B. B is the largest such n \u2264 300000. A is the count.\n\nI need the largest triplet (p,p+4,p+6) with p \u2264 300000.\n\nLet me search near 300000 downward for such triplets. We need p prime, p+4 prime, p+6 prime, p\u22611 mod3.\n\nLet me check primes near 300000.\n\n299999? Let's find. Actually let me look for p just below 300000.\n\nThe known largest term of A022004 below 300000... Let me try to recall or compute.\n\nLet me search candidates. n \u2261 1 mod 3 and n\u22612 or 3 mod 5.\n\nLet me check around 299000-300000.\n\nLet me test some values. I'll check p, p+4, p+6.\n\nConsider p = 299797? Hard without primality.\n\nThis manual approach is impractical. Let me think about what a reasonable intended answer is.\n\nActually, let me reconsider\u2014maybe I should just carefully compute using known data. The count of prime triplets (p,p+4,p+6) below 300000.\n\nHmm. Let me try to recall the count. The number of prime triplets of both types below various limits... \n\nActually, let me just try to determine B by checking primes near 300000.\n\nLet me list primes just below 300000: 299999 (is it prime? 299999 = 300000-1. 299999 / 7 = 42857, 7*42857=299999. Yes! So 299999=7*42857, not prime).\n\nLet me find the actual triplet. Let me look for the largest p.\n\nLet me try to work with candidates systematically near the top.\n\nPrimes near 300000: 299993 is prime (I think). Let's check 299993, 299997, 299999. \n299993+4=299997, 299997 divisible by 3 (2+9+9+9+9+7=45, yes). So not.\n\nWe need n\u22611 mod3. n=299993: 2+9+9+9+9+3=41, 41 mod3=2, so n\u22612 mod3. Not valid form (we need n\u22611 mod3).\n\nLet me find n\u22611 mod3 near 300000, prime.\n300000 mod 3 =0. So n\u22611 mod3 means n\u22611 mod3, e.g., 299998? even. n odd and \u22611 mod3: numbers \u22611 mod3 that are odd: 300001 too big. Below: 299995 (odd, mod3: 2+9+9+9+9+5=43\u22611). 299995\u22611 mod3. Divisible by 5 (ends in 5). Not prime.\nNext lower \u22611 mod3 odd: 299989 (43-6... let's compute: 299989 digit sum=2+9+9+9+8+9=46\u22611 mod3). 299989 odd. Is it prime? Also need mod5: ends in 9, ok. Then n+4=299993, n+6=299995(div by5, not prime). So fails at n+6.\n\nWe also need n\u22612 or 3 mod5 (from earlier). 299989 mod5 = 4 (ends 9). n\u22614 means n+6\u22610 mod5. Indeed 299995 div by 5. Fails.\n\nSo need n\u22612 or3 mod5. n\u22611 mod3. Let's find candidates.\n\nn\u22611 mod3, n\u22612 mod5 or n\u22613 mod5.\n\nn\u22611 mod3, n\u22612 mod5: CRT \u2192 n\u22617 mod15? Check 7: 7mod3=1,7mod5=2. Yes n\u22617 mod15.\nn\u22611 mod3, n\u22613 mod5: n\u226113 mod15? 13mod3=1,13mod5=3. Yes n\u226113 mod15.\n\nSo n\u22617 or 13 mod15.\n\nNear 300000: 300000 mod15=0. So n\u22617 mod15 \u2192 n=299992+? 300000-15k+7... 299992? 299992 mod15: 300000-8=299992, 299992 mod15 = -8 mod15=7. So 299992\u22617 mod15, but even. Need odd. The residue mod15 with parity alternates. n\u22617 mod15: values 299992(even),299977(odd),299962(even),299947(odd)...\n\nn\u226113 mod15: 299998(even? 299998 mod15: 300000-2=299998\u226113. even),299983(odd),299968(even),299953(odd)...\n\nOdd candidates descending: \n299983 (\u226113 mod15), 299977(\u22617), 299953(\u226113), 299947(\u22617), 299923(\u226113), 299917(\u22617),...\n\nLet me test 299983: n=299983, n+4=299987, n+6=299989.\nCheck primality of each.\n299983: is it prime? Let me test small divisors.\n299989 we considered. Let me just check these three.\n\n299983: sqrt \u2248 547.7. Need to check primes up to 547.\nDiv by 7: 7*42854=299978, remainder5. No.\n11: alternating sum: 3-8+9-9+9-2? Let me do 299983: digits 2,9,9,9,8,3. Alt sum from right: 3-8+9-9+9-2=2. Not div by 11.\n13: 299983/13=23075.6..., 13*23075=299975, rem8. No.\n17: 17*17646=299982, rem1. No.\n19: 19*15788=299972, rem11. No.\n23: 23*13043=300, - let's 23*13042=299966, rem17. No.\n29: 29*10344=299976, rem7. No.\n31: 31*9677=299987, too big; 31*9676=299956, rem27. No.\n37: 37*8107=299959, rem24. No.\n41: 41*7316=299956, rem27. No.\n43: 43*6976=299968, rem15. No.\n47: 47*6382=299954, rem29. No.\n53: 53*5660=299980, rem3. No.\n59: 59*5084=299956, rem27. No.\n61: 61*4917=299937, rem46. No.\n67: 67*4477=299959, rem24. No.\n71: 71*4225=299975, rem8. No.\n73: 73*4109=299957, rem26. No.\n79: 79*3797=299963, rem20. No.\n83: 83*3614=299962, rem21. No.\n89: 89*3370=299930, rem53. No.\n97: 97*3092=299924, rem59. No.\n101: 101*2970=299970, rem13. No.\n103: 103*2912=299936, rem47. No.\n107: 107*2803=299921, rem62. No.\n109: 109*2752=299968, rem15. No.\n113: 113*2654=299902, rem81. No.\n127: 127*2362=299974, rem9. No.\n131: 131*2290=299990, too big; 131*2289=299859, rem124. No.\n137: 137*2189=299893, rem90. No.\n139: 139*2158=299962, rem21. No.\n149: 149*2013=299937, rem46. No.\n151: 151*1986=299886, rem97. No.\n157: 157*1910=299870, rem113. No.\n163: 163*1840=299920, rem63. No.\n167: 167*1796=299932, rem51. No.\n173: 173*1734=299982, rem1. No.\n179: 179*1675=299825, rem158. No.\n181: 181*1657=299917, rem66. No.\n191: 191*1570=299870, rem113. No.\n193: 193*1554=299922, rem61. No.\n197: 197*1522=299834, rem149. No.\n199: 199*1507=299893, rem90. No.\n211: 211*1421=299831, rem152. No.\n223: 223*1345=299935, rem48. No.\n227: 227*1321=299867, rem116. No.\n229: 229*1310=299990, too big; 229*1309=299761, rem222. No.\n233: 233*1287=299871, rem112. No.\n239: 239*1255=299945, rem38. No.\n241: 241*1244=299804, rem179. No.\n251: 251*1195=299945, rem38. No.\n257: 257*1167=299919, rem64. No.\n263: 263*1140=299820, rem163. No.\n269: 269*1115=299935, rem48. No.\n271: 271*1106=299726, rem257. No.\n277: 277*1083=299991, too big; 277*1082=299714, rem269. No.\n281: 281*1067=299827, rem156. No.\n283: 283*1060=299980, rem3. No.\n293: 293*1023=299739, rem244. No.\n307: 307*977=299939, rem44. No.\n311: 311*964=299804, rem179. No.\n313: 313*958=299854, rem129. No.\n317: 317*946=299882, rem101. No.\n331: 331*906=299886, rem97. No.\n337: 337*890=299930, rem53. No.\n347: 347*864=299808, rem175. No.\n349: 349*859=299791, rem192. No.\n353: 353*849=299697, rem286. No.\n359: 359*835=299765, rem218. No.\n367: 367*817=299839, rem144. No.\n373: 373*804=299892, rem91. No.\n379: 379*791=299789, rem194. No.\n383: 383*783=299889, rem94. No.\n389: 389*771=299919, rem64. No.\n397: 397*755=299735, rem248. No.\n401: 401*748=299948, rem35. No.\n409: 409*733=299797, rem186. No.\n419: 419*716=299 - 419*716=299,   419*716=299944, rem39. No.\n421: 421*712=299752, rem231. No.\n431: 431*696=299976, rem7. No.\n433: 433*692=299636, rem347. No.\n439: 439*683=299837, rem146. No.\n443: 443*677=299911, rem72. No.\n449: 449*668=299932, rem51. No.\n457: 457*656=299792, rem191. No.\n461: 461*650=299650, rem333. No.\n463: 463*647=299561, rem422. No.\n467: 467*642=299814, rem169. No.\n479: 479*626=299854, rem129. No.\n487: 487*616=299992, too big; 487*615=299505, rem478. No.\n491: 491*611=300001, too big; 491*610=299510, rem473. No.\n499: 499*601=299899, rem84. No.\n503: 503*596=299788, rem195. No.\n509: 509*589=299801, rem182. No.\n521: 521*575=299575, rem408. No.\n523: 523*573=299679, rem304. No.\n541: 541*554=299714, rem269. No.\n547: 547*548=299756, rem227. No.\n\nSo 299983 is prime! \n\nNow n+4=299987. Check primality. sqrt~547.7.\n299987: \n7: 7*42855=299985, rem2. No.\n11: digits 2,9,9,9,8,7 alt sum from right:7-8+9-9+9-2=6. No.\n13: 13*23076=299988, too big; 13*23075=299975, rem12. No.\n17: 17*17646=299982, rem5. No.\n19: 19*15788=299972, rem15. No.\n23: 23*13042=299966, rem21. No.\n29: 29*10344=299976, rem11. No.\n31: 31*9677=299987! Let me check: 31*9677 = 31*9000=279000, 31*677=20987, total 299987. Yes! So 299987=31*9677. Not prime.\n\nSo n=299983 fails (n+4 not prime).\n\nNext candidate down: 299977 (\u22617 mod15). n=299977,n+4=299981,n+6=299983.\nWe know 299983 is prime. Need 299977 and 299981 prime.\n299977: \n7: 7*42854=299978, so rem -1 \u2192299977=7*42853+6, rem6. No.\n11: digits 2,9,9,9,7,7 alt from right:7-7+9-9+9-2=7. No.\n13: 13*23075=299975, rem2. No.\n17: 17*17645=299965, rem12. No.\n19: 19*15788=299972, rem5. No.\n23: 23*13042=299966, rem11. No.\n29: 29*10344=299976, rem1. No.\n31: 31*9676=299956, rem21. No.\n37: 37*8107=299959, rem18. No.\n41: 41*7316=299956, rem21. No.\n43: 43*6976=299968, rem9. No.\n47: 47*6382=299954, rem23. No.\n53: 53*5660=299980, too big; 53*5659=299927, rem50. No.\n59: 59*5084=299956, rem21. No.\n61: 61*4917=299937, rem40. No.\n67: 67*4477=299959, rem18. No.\n71: 71*4225=299975, rem2. No.\n73: 73*4109=299957, rem20. No.\n79: 79*3797=299963, rem14. No.\n83: 83*3614=299962, rem15. No.\n89: 89*3370=299930, rem47. No.\n97: 97*3092=299924, rem53. No.\n101: 101*2970=299970, rem7. No.\n103: 103*2912=299936, rem41. No.\n107: 107*2803=299921, rem56. No.\n109: 109*2752=299968, rem9. No.\n113: 113*2654=299902, rem75. No.\n127: 127*2362=299974, rem3. No.\n131: 131*2289=299859, rem118. No.\n137: 137*2189=299893, rem84. No.\n139: 139*2158=299962, rem15. No.\n149: 149*2013=299937, rem40. No.\n151: 151*1986=299886, rem91. No.\n157: 157*1910=299870, rem107. No.\n163: 163*1840=299920, rem57. No.\n167: 167*1796=299932, rem45. No.\n173: 173*1734=299982, too big; 173*1733=299809, rem168. No.\n179: 179*1675=299825, rem152. No.\n181: 181*1657=299917, rem60. No.\n191: 191*1570=299870, rem107. No.\n193: 193*1554=299922, rem55. No.\n197: 197*1522=299834, rem143. No.\n199: 199*1507=299893, rem84. No.\n211: 211*1421=299831, rem146. No.\n223: 223*1345=299935, rem42. No.\n227: 227*1321=299867, rem110. No.\n229: 229*1309=299761, rem216. No.\n233: 233*1287=299871, rem106. No.\n239: 239*1255=299945, rem32. No.\n241: 241*1244=299804, rem173. No.\n251: 251*1195=299945, rem32. No.\n257: 257*1167=299919, rem58. No.\n263: 263*1140=299820, rem157. No.\n269: 269*1114=299666, rem311\u2192 too big rem, recompute 269*1115=299935, rem42. No.\n271: 271*1106=299726, rem251. No.\n277: 277*1082=299714, rem263. No.\n281: 281*1067=299827, rem150. No.\n283: 283*1060=299980, too big; 283*1059=299697, rem280. No.\n293: 293*1023=299739, rem238. No.\n307: 307*977=299939, rem38. No.\n311: 311*964=299804, rem173. No.\n313: 313*958=299854, rem123. No.\n317: 317*946=299882, rem95. No.\n331: 331*906=299886, rem91. No.\n337: 337*890=299930, rem47. No.\n347: 347*864=299808, rem169. No.\n349: 349*859=299791, rem186. No.\n353: 353*849=299697, rem280. No.\n359: 359*835=299765, rem212. No.\n367: 367*817=299839, rem138. No.\n373: 373*804=299892, rem85. No.\n379: 379*791=299789, rem188. No.\n383: 383*783=299889, rem88. No.\n389: 389*771=299919, rem58. No.\n397: 397*755=299735, rem242. No.\n401: 401*748=299948, rem29. No.\n409: 409*733=299797, rem180. No.\n419: 419*716=299944, rem33. No.\n421: 421*712=299752, rem225. No.\n431: 431*696=299976, rem1. No.\n433: 433*692=299636, rem341. No.\n439: 439*683=299837, rem140. No.\n443: 443*677=299911, rem66. No.\n449: 449*668=299932, rem45. No.\n457: 457*656=299792, rem185. No.\n461: 461*650=299650, rem327. No.\n463: 463*647=299561, rem416. No.\n467: 467*642=299814, rem163. No.\n479: 479*626=299854, rem123. No.\n487: 487*615=299505, rem472. No.\n491: 491*610=299510, rem467. No.\n499: 499*601=299899, rem78. No.\n503: 503*596=299788, rem189. No.\n509: 509*589=299801, rem176. No.\n521: 521*575=299575, rem402. No.\n523: 523*573=299679, rem298. No.\n541: 541*554=299714, rem263. No.\n547: 547*548=299756, rem221. No.\n\nSo 299977 is prime.\n\nNow 299981 (=n+4). Check primality.\n299981:\n7: 7*42854=299978, rem3. No.\n11: digits 2,9,9,9,8,1 alt from right:1-8+9-9+9-2=0. Divisible by 11! So 299981=11*27271. Let me verify: 11*27271=299981. Yes. Not prime.\n\nSo 299977 fails (n+4 not prime).\n\nNext candidate: 299953 (\u226113 mod15). n=299953,n+4=299957,n+6=299959.\nCheck each.\n299953:\n7: 7*42850=299950, rem3. No.\n11: digits2,9,9,9,5,3 alt:3-5+9-9+9-2=5. No.\n13: 13*23073=299949, rem4. No.\n17: 17*17644=299948, rem5. No.\n19: 19*15787=299953! Check:19*15787=19*15000=285000,19*787=14953, total 299953. Yes! Divisible by 19. Not prime.\n\nFails.\n\nNext: 299947 (\u22617 mod15). n=299947,n+4=299951,n+6=299953.\nBut 299953 is composite (=19*15787), so n+6 composite. Fails.\n\nNext: 299923 (\u226113 mod15). n=299923,n+4=299927,n+6=299929.\nCheck 299923:\n7: 7*42846=299922, rem1. No.\n11: digits2,9,9,9,2,3 alt:3-2+9-9+9-2=8. No.\n13: 13*23071=299923? 13*23071=299923. Check:13*23000=299000,13*71=923, total299923. Yes! Divisible by 13. Not prime.\n\nFails.\n\nNext: 299917 (\u22617 mod15). n=299917,n+4=299921,n+6=299923.\n299923 composite. Fails.\n\nNext: 299893 (\u226113 mod15). n=299893, n+4=299897,n+6=299899.\nCheck 299893:\n7:7*42841=299887,rem6. No.\n11: digits2,9,9,8,9,3 alt:3-9+8-9+9-2=0. Divisible by 11! 299893=11*27263. Not prime.\n\nFails.\n\nNext: 299887(\u22617 mod15). n=299887,n+4=299891,n+6=299893.\n299893 composite. Fails.\n\nNext:299863(\u226113 mod15). n=299863,n+4=299867,n+6=299869.\nCheck 299863:\n7:7*42837=299859,rem4. No.\n11:digits2,9,9,8,6,3 alt:3-6+8-9+9-2=3. No.\n13:13*23066=299858,rem5. No.\n17:17*17639=299863? 17*17639=17*17000=289000,17*639=10863,total299863. Yes! Divisible by17. Not prime.\n\nFails.\n\nNext:299857(\u22617 mod15). n=299857,n+4=299861,n+6=299863.\n299863 composite. Fails.\n\nNext:299833(\u226113 mod15). n=299833,n+4=299837,n+6=299839.\nCheck 299833:\n7:7*42833=299831,rem2. No.\n11:digits2,9,9,8,3,3 alt:3-3+8-9+9-2=6. No.\n13:13*23064=299832,rem1. No.\n17:17*17637=299829,rem4. No.\n19:19*15780=299820,rem13. No.\n23:23*13036=299828,rem5. No.\n29:29*10339=299831,rem2. No.\n31:31*9672=299832,rem1. No.\n37:37*8103=299811,rem22. No.\n41:41*7313=299833? 41*7313=41*7000=287000,41*313=12833,total299833. Yes! Divisible by41. Not prime.\n\nFails.\n\nNext:299827(\u22617 mod15). n=299827,n+4=299831,n+6=299833.\n299833 composite. Fails.\n\nNext:299803(\u226113 mod15). n=299803,n+4=299807,n+6=299809.\nCheck299803:\n7:7*42829=299803? 7*42829=299803. 7*42800=299600,7*29=203,total299803. Yes! Divisible by7. Not prime.\n\nFails.\n\nNext:299797(\u22617 mod15). n=299797,n+4=299801,n+6=299803.\n299803 composite. Fails.\n\nNext:299773(\u226113 mod15). n=299773,n+4=299777,n+6=299779.\nCheck299773:\n7:7*42824=299768,rem5. No.\n11:digits2,9,9,7,7,3 alt:3-7+7-9+9-2=1. No.\n13:13*23059=299767,rem6. No.\n17:17*17633=299761,rem12. No.\n19:19*15777=299763,rem10. No.\n23:23*13033=299759,rem14. No.\n29:29*10337=299773? 29*10337=29*10000=290000,29*337=9773,total299773. Yes! Divisible by29. Not prime.\n\nFails.\n\nNext:299767(\u22617 mod15). n=299767,n+4=299771,n+6=299773.\n299773 composite. Fails.\n\nNext:299743(\u226113 mod15). n=299743,n+4=299747,n+6=299749.\nCheck299743:\n7:7*42820=299740,rem3. No.\n11:digits2,9,9,7,4,3 alt:3-4+7-9+9-2=4. No.\n13:13*23057=299741,rem2. No.\n17:17*17632=299744,too big;17*17631=299727,rem16. No.\n19:19*15776=299744,too big;19*15775=299725,rem18. No.\n23:23*13032=299736,rem7. No.\n29:29*10336=299744,too big;29*10335=299715,rem28. No.\n31:31*9669=299739,rem4. No.\n37:37*8101=299737,rem6. No.\n41:41*7310=299710,rem33. No.\n43:43*6971=299753,too big;43*6970=299710,rem33. No.\n47:47*6377=299719,rem24. No.\n53:53*5655=299715,rem28. No.\n59:59*5080=299720,rem23. No.\n61:61*4913=299693,rem50. No.\n67:67*4473=299691,rem52. No.\n71:71*4221=299691,rem52. No.\n73:73*4106=299738,rem5. No.\n79:79*3794=299726,rem17. No.\n83:83*3611=299713,rem30. No.\n89:89*3367=299663,rem80. No.\n97:97*3090=299730,rem13. No.\n101:101*2967=299667,rem76. No.\n103:103*2910=299730,rem13. No.\n107:107*2801=299707,rem36. No.\n109:109*2749=299641,rem102. No.\n113:113*2652=299676,rem67. No.\n127:127*2360=299720,rem23. No.\n131:131*2288=299728,rem15. No.\n137:137*2188=299756,too big;137*2187=299619,rem124. No.\n139:139*2156=299684,rem59. No.\n149:149*2011=299639,rem104. No.\n151:151*1985=299735,rem8. No.\n157:157*1909=299713,rem30. No.\n163:163*1839=299757,too big;163*1838=299594,rem149. No.\n167:167*1794=299598,rem145. No.\n173:173*1732=299636,rem107. No.\n179:179*1674=299646,rem97. No.\n181:181*1656=299736,rem7. No.\n191:191*1569=299679,rem64. No.\n193:193*1553=299729,rem14. No.\n197:197*1521=299637,rem106. No.\n199:199*1506=299694,rem49. No.\n211:211*1420=299620,rem123. No.\n223:223*1344=299712,rem31. No.\n227:227*1320=299640,rem103. No.\n229:229*1309=299761,too big;229*1308=299532,rem211. No.\n233:233*1286=299638,rem105. No.\n239:239*1254=299706,rem37. No.\n241:241*1243=299563,rem180. No.\n251:251*1194=299694,rem49. No.\n257:257*1166=299662,rem81. No.\n263:263*1139=299557,rem186. No.\n269:269*1114=299666,rem77. No.\n271:271*1106=299726,rem17. No.\n277:277*1082=299714,rem29. No.\n281:281*1066=299546,rem197. No.\n283:283*1059=299697,rem46. No.\n293:293*1023=299739,rem4. No.\n307:307*976=299632,rem111. No.\n311:311*964=299804,too big;311*963=299493,rem250. No.\n313:313*957=299541,rem202. No.\n317:317*945=299565,rem178. No.\n331:331*905=299555,rem188. No.\n337:337*889=299593,rem150. No.\n347:347*864=299808,too big;347*863=299461,rem282. No.\n349:349*859=299791,too big;349*858=299442,rem301. No.\n353:353*849=299697,rem46. No.\n359:359*835=299765,too big;359*834=299406,rem337. No.\n367:367*816=299472,rem271. No.\n373:373*803=299519,rem224. No.\n379:379*791=299789,too big;379*790=299410,rem333. No.\n383:383*782=299506,rem237. No.\n389:389*770=299530,rem213. No.\n397:397*755=299735,rem8. No.\n401:401*747=299547,rem196. No.\n409:409*732=299388,rem355. No.\n419:419*715=299585,rem158. No.\n421:421*712=299752,too big;421*711=299331,rem412. No.\n431:431*695=299545,rem198. No.\n433:433*692=299636,rem107. No.\n439:439*682=299398,rem345. No.\n443:443*676=299468,rem275. No.\n449:449*667=299483,rem260. No.\n457:457*656=299792,too big;457*655=299335,rem408. No.\n461:461*650=299650,rem93. No.\n463:463*647=299561,rem182. No.\n467:467*642=299814,too big;467*641=299347,rem396. No.\n479:479*625=299375,rem368. No.\n487:487*615=299505,rem238. No.\n491:491*610=299510,rem233. No.\n499:499*600=299400,rem343. No.\n503:503*596=299788,too big;503*595=299285,rem458. No.\n509:509*589=299801,too big;509*588=299292,rem451. No.\n521:521*575=299575,rem168. No.\n523:523*573=299679,rem64. No.\n541:541*554=299714,rem29. No.\n547:547*547=299209,rem534. Actually 547*548=299756 too big, 547*547=299209, rem534. Hmm 534<547 ok. No.\n\nSo 299743 is prime.\n\nNow n+4=299747. Check.\n299747:\n7:7*42821=299747? 7*42821=299747. 7*42800=299600,7*21=147,total299747. Yes! Divisible by7. Not prime.\n\nFails.\n\nNext:299737(\u22617 mod15). n=299737,n+4=299741,n+6=299743.\n299743 is prime (n+6). Need 299737 and 299741 prime.\n299737:\n7:7*42819=299733,rem4. No.\n11:digits2,9,9,7,3,7 alt:7-3+7-9+9-2=9. No.\n13:13*23056=299728,rem9. No.\n17:17*17631=299727,rem10. No.\n19:19*15775=299725,rem12. No.\n23:23*13032=299736,rem1. No.\n29:29*10335=299715,rem22. No.\n31:31*9668=299708,rem29. No.\n37:37*8101=299737? 37*8101=37*8000=296000,37*101=3737,total299737. Yes! Divisible by37. Not prime.\n\nFails.\n\nNext:299713(\u226113 mod15). n=299713,n+4=299717,n+6=299719.\n299713:\n7:7*42816=299712,rem1. No.\n11:digits2,9,9,7,1,3 alt:3-1+7-9+9-2=7. No.\n13:13*23054=299702,rem11. No.\n17:17*17630=299710,rem3. No.\n19:19*15774=299706,rem7. No.\n23:23*13031=299713? 23*13031=23*13000=299000,23*31=713,total299713. Yes! Divisible by23. Not prime.\n\nFails.\n\nNext:299707(\u22617 mod15). n=299707,n+4=299711,n+6=299713.\n299713 composite. Fails.\n\nNext:299683(\u226113 mod15). n=299683,n+4=299687,n+6=299689.\n299683:\n7:7*42811=299677,rem6. No.\n11:digits2,9,9,6,8,3 alt:3-8+6-9+9-2=-1. No.\n13:13*23052=299676,rem7. No.\n17:17*17628=299676,rem7. No.\n19:19*15772=299668,rem15. No.\n23:23*13029=299667,rem16. No.\n29:29*10334=299686,too big;29*10333=299657,rem26. No.\n31:31*9667=299677,rem6. No.\n37:37*8099=299663,rem20. No.\n41:41*7309=299669,rem14. No.\n43:43*6969=299667,rem16. No.\n47:47*6376=299672,rem11. No.\n53:53*5654=299662,rem21. No.\n59:59*5079=299661,rem22. No.\n61:61*4912=299632,rem51. No.\n67:67*4472=299624,rem59. No.\n71:71*4221=299691,too big;71*4220=299620,rem63. No.\n73:73*4105=299665,rem18. No.\n79:79*3793=299647,rem36. No.\n83:83*3610=299630,rem53. No.\n89:89*3367=299663,rem20. No.\n97:97*3089=299633,rem50. No.\n101:101*2967=299667,rem16. No.\n103:103*2909=299627,rem56. No.\n107:107*2800=299600,rem83. No.\n109:109*2749=299641,rem42. No.\n113:113*2652=299676,rem7. No.\n127:127*2359=299593,rem90. No.\n131:131*2287=299597,rem86. No.\n137:137*2187=299619,rem64. No.\n139:139*2155=299545,rem138. No.\n149:149*2011=299639,rem44. No.\n151:151*1984=299584,rem99. No.\n157:157*1908=299556,rem127. No.\n163:163*1838=299594,rem89. No.\n167:167*1794=299598,rem85. No.\n173:173*1732=299636,rem47. No.\n179:179*1674=299646,rem37. No.\n181:181*1655=299555,rem128. No.\n191:191*1568=299488,rem195\u2192195>191, recompute:191*1569=299679,rem4. No.\n193:193*1552=299536,rem147. No.\n197:197*1521=299637,rem46. No.\n199:199*1505=299495,rem188. No.\n211:211*1420=299620,rem63. No.\n223:223*1343=299489,rem194. No.\n227:227*1320=299640,rem43. No.\n229:229*1308=299532,rem151. No.\n233:233*1286=299638,rem45. No.\n239:239*1253=299467,rem216. No.\n241:241*1243=299563,rem120. No.\n251:251*1193=299443,rem240. No.\n257:257*1166=299662,rem21. No.\n263:263*1139=299557,rem126. No.\n269:269*1113=299397,rem286\u2192286>269, recompute 269*1114=299666,rem17. No.\n271:271*1105=299455,rem228. No.\n277:277*1081=299437,rem246. No.\n281:281*1066=299546,rem137. No.\n283:283*1059=299697,too big;283*1058=299414,rem269. No.\n293:293*1022=299446,rem237. No.\n307:307*976=299632,rem51. No.\n311:311*963=299493,rem190. No.\n313:313*957=299541,rem142. No.\n317:317*945=299565,rem118. No.\n331:331*905=299555,rem128. No.\n337:337*889=299593,rem90. No.\n347:347*863=299461,rem222. No.\n349:349*858=299442,rem241. No.\n353:353*849=299697,too big;353*848=299344,rem339. No.\n359:359*834=299406,rem277. No.\n367:367*816=299472,rem211. No.\n373:373*803=299519,rem164. No.\n379:379*790=299410,rem273. No.\n383:383*782=299506,rem177. No.\n389:389*770=299530,rem153. No.\n397:397*754=299338,rem345. No.\n401:401*747=299547,rem136. No.\n409:409*732=299388,rem295. No.\n419:419*715=299585,rem98. No.\n421:421*711=299331,rem352. No.\n431:431*695=299545,rem138. No.\n433:433*692=299636,rem47. No.\n439:439*682=299398,rem285. No.\n443:443*676=299468,rem215. No.\n449:449*667=299483,rem200. No.\n457:457*655=299335,rem348. No.\n461:461*650=299650,rem33. No.\n463:463*647=299561,rem122. No.\n467:467*641=299347,rem336. No.\n479:479*625=299375,rem308. No.\n487:487*615=299505,rem178. No.\n491:491*610=299510,rem173. No.\n499:499*600=299400,rem283. No.\n503:503*595=299285,rem398. No.\n509:509*588=299292,rem391. No.\n521:521*575=299575,rem108. No.\n523:523*573=299679,rem4. No.\n541:541*554=299714,too big;541*553=299173,rem510. No.\n547:547*547=299209,rem474. No.\n\nSo 299683 is prime.\n\nn+4=299687. Check.\n299687:\n7:7*42812=299684,rem3. No.\n11:digits2,9,9,6,8,7 alt:7-8+6-9+9-2=3. No.\n13:13*23052=299676,rem11. No.\n17:17*17628=299676,rem11. No.\n19:19*15773=299687? 19*15773=19*15000=285000,19*773=14687,total299687. Yes! Divisible by19. Not prime.\n\nFails.\n\nNext:299677(\u22617 mod15). n=299677,n+4=299681,n+6=299683.\n299683 prime. Need 299677 and 299681 prime.\n299677:\n7:7*42811=299677? 7*42811=299677. 7*42800=299600,7*11=77,total299677. Yes! Divisible by7. Not prime.\n\nFails.\n\nNext:299653(\u226113 mod15). n=299653,n+4=299657,n+6=299659.\n299653:\n7:7*42807=299649,rem4. No.\n11:digits2,9,9,6,5,3 alt:3-5+6-9+9-2=2. No.\n13:13*23050=299650,rem3. No.\n17:17*17626=299642,rem11. No.\n19:19*15771=299649,rem4. No.\n23:23*13028=299644,rem9. No.\n29:29*10333=299657,too big;29*10332=299628,rem25. No.\n31:31*9666=299646,rem7. No.\n37:37*8098=299626,rem27. No.\n41:41*7308=299628,rem25. No.\n43:43*6968=299624,rem29. No.\n47:47*6375=299625,rem28. No.\n53:53*5653=299609,rem44. No.\n59:59*5078=299602,rem51. No.\n61:61*4912=299632,rem21. No.\n67:67*4472=299624,rem29. No.\n71:71*4220=299620,rem33. No.\n73:73*4104=299592,rem61. No.\n79:79*3792=299568,rem85\u219285>79, recompute 79*3793=299647,rem6. No.\n83:83*3610=299630,rem23. No.\n89:89*3366=299574,rem79. No.\n97:97*3089=299633,rem20. No.\n101:101*2966=299566,rem87. No.\n103:103*2909=299627,rem26. No.\n107:107*2800=299600,rem53. No.\n109:109*2749=299641,rem12. No.\n113:113*2652=299676,too big;113*2651=299563,rem90. No.\n127:127*2359=299593,rem60. No.\n131:131*2287=299597,rem56. No.\n137:137*2187=299619,rem34. No.\n139:139*2155=299545,rem108. No.\n149:149*2011=299639,rem14. No.\n151:151*1984=299584,rem69. No.\n157:157*1908=299556,rem97. No.\n163:163*1838=299594,rem59. No.\n167:167*1794=299598,rem55. No.\n173:173*1732=299636,rem17. No.\n179:179*1674=299646,rem7. No.\n181:181*1655=299555,rem98. No.\n191:191*1569=299679,too big;191*1568=299488,rem165. No.\n193:193*1552=299536,rem117. No.\n197:197*1521=299637,rem16. No.\n199:199*1505=299495,rem158. No.\n211:211*1420=299620,rem33. No.\n223:223*1343=299489,rem164. No.\n227:227*1320=299640,rem13. No.\n229:229*1308=299532,rem121. No.\n233:233*1286=299638,rem15. No.\n239:239*1253=299467,rem186. No.\n241:241*1243=299563,rem90. No.\n251:251*1193=299443,rem210. No.\n257:257*1166=299662,too big;257*1165=299405,rem248. No.\n263:263*1139=299557,rem96. No.\n269:269*1114=299666,too big;269*1113=299397,rem256. No.\n271:271*1105=299455,rem198. No.\n277:277*1081=299437,rem216. No.\n281:281*1066=299546,rem107. No.\n283:283*1058=299414,rem239. No.\n293:293*1022=299446,rem207. No.\n307:307*976=299632,rem21. No.\n311:311*963=299493,rem160. No.\n313:313*957=299541,rem112. No.\n317:317*945=299565,rem88. No.\n331:331*905=299555,rem98. No.\n337:337*889=299593,rem60. No.\n347:347*863=299461,rem192. No.\n349:349*858=299442,rem211. No.\n353:353*848=299344,rem309. No.\n359:359*834=299406,rem247. No.\n367:367*816=299472,rem181. No.\n373:373*803=299519,rem134. No.\n379:379*790=299410,rem243. No.\n383:383*782=299506,rem147. No.\n389:389*770=299530,rem123. No.\n397:397*754=299338,rem315. No.\n401:401*747=299547,rem106. No.\n409:409*732=299388,rem265. No.\n419:419*715=299585,rem68. No.\n421:421*711=299331,rem322. No.\n431:431*695=299545,rem108. No.\n433:433*692=299636,rem17. No.\n439:439*682=299398,rem255. No.\n443:443*676=299468,rem185. No.\n449:449*667=299483,rem170. No.\n457:457*655=299335,rem318. No.\n461:461*650=299650,rem3. No.\n463:463*647=299561,rem92. No.\n467:467*641=299347,rem306. No.\n479:479*625=299375,rem278. No.\n487:487*615=299505,rem148. No.\n491:491*610=299510,rem143. No.\n499:499*600=299400,rem253. No.\n503:503*595=299285,rem368. No.\n509:509*588=299292,rem361. No.\n521:521*575=299575,rem78. No.\n523:523*572=299156,rem497\u2192 recompute 523*573=299679,too big; 523*572=299156, rem497>523? No 497<523. So rem497. No.\n541:541*553=299173,rem480. No.\n547:547*547=299209,rem444. No.\n\nSo 299653 is prime.\n\nn+4=299657. Check.\n299657:\n7:7*42808=299656,rem1. No.\n11:digits2,9,9,6,5,7 alt:7-5+6-9+9-2=6. No.\n13:13*23050=299650,rem7. No.\n17:17*17627=299659,too big;17*17626=299642,rem15. No.\n19:19*15771=299649,rem8. No.\n23:23*13028=299644,rem13. No.\n29:29*10333=299657? 29*10333=29*10000=290000,29*333=9657,total299657. Yes! Divisible by29. Not prime.\n\nFails.\n\nNext:299647(\u22617 mod15). n=299647,n+4=299651,n+6=299653.\n299653 prime. Need 299647,299651 prime.\n299647:\n7:7*42806=299642,rem5. No.\n11:digits2,9,9,6,4,7 alt:7-4+6-9+9-2=7. No.\n13:13*23049=299637,rem10. No.\n17:17*17626=299642,rem5. No.\n19:19*15770=299630,rem17. No.\n23:23*13028=299644,rem3. No.\n29:29*10332=299628,rem19. No.\n31:31*9666=299646,rem1. No.\n37:37*8098=299626,rem21. No.\n41:41*7308=299628,rem19. No.\n43:43*6968=299624,rem23. No.\n47:47*6375=299625,rem22. No.\n53:53*5653=299609,rem38. No.\n59:59*5078=299602,rem45. No.\n61:61*4912=299632,rem15. No.\n67:67*4472=299624,rem23. No.\n71:71*4220=299620,rem27. No.\n73:73*4104=299592,rem55. No.\n79:79*3793=299647? 79*3793=79*3800=300200 minus79*7=553 \u2192300200-553=299647. Yes! Divisible by79. Not prime.\n\nFails.\n\nNext:299623(\u226113 mod15). n=299623,n+4=299627,n+6=299629.\n299623:\n7:7*42803=299621,rem2. No.\n11:digits2,9,9,6,2,3 alt:3-2+6-9+9-2=5. No.\n13:13*23048=299624,too big;13*23047=299611,rem12. No.\n17:17*17624=299608,rem15. No.\n19:19*15769=299611,rem12. No.\n23:23*13027=299621,rem2. No.\n29:29*10331=299599,rem24. No.\n31:31*9665=299615,rem8. No.\n37:37*8098=299626,too big;37*8097=299589,rem34. No.\n41:41*7307=299587,rem36. No.\n43:43*6968=299624,too big;43*6967=299581,rem42. No.\n47:47*6375=299625,too big;47*6374=299578,rem45. No.\n53:53*5653=299609,rem14. No.\n59:59*5078=299602,rem21. No.\n61:61*4912=299632,too big;61*4911=299571,rem52. No.\n67:67*4472=299624,too big;67*4471=299557,rem66. No.\n71:71*4220=299620,rem3. No.\n73:73*4104=299592,rem31. No.\n79:79*3792=299568,rem55. No.\n83:83*3610=299630,too big;83*3609=299547,rem76. No.\n89:89*3366=299574,rem49. No.\n97:97*3089=299633,too big;97*3088=299536,rem87. No.\n101:101*2966=299566,rem57. No.\n103:103*2909=299627,too big;103*2908=299524,rem99. No.\n107:107*2800=299600,rem23. No.\n109:109*2748=299532,rem91. No.\n113:113*2651=299563,rem60. No.\n127:127*2359=299593,rem30. No.\n131:131*2287=299597,rem26. No.\n137:137*2186=299482,rem141\u2192141>137,recompute 137*2187=299619,rem4. No.\n139:139*2155=299545,rem78. No.\n149:149*2011=299639,too big;149*2010=299490,rem133. No.\n151:151*1984=299584,rem39. No.\n157:157*1908=299556,rem67. No.\n163:163*1838=299594,rem29. No.\n167:167*1794=299598,rem25. No.\n173:173*1732=299636,too big;173*1731=299463,rem160. No.\n179:179*1674=299646,too big;179*1673=299467,rem156. No.\n181:181*1655=299555,rem68. No.\n191:191*1568=299488,rem135. No.\n193:193*1552=299536,rem87. No.\n197:197*1521=299637,too big;197*1520=299440,rem183. No.\n199:199*1505=299495,rem128. No.\n211:211*1420=299620,rem3. No.\n223:223*1343=299489,rem134. No.\n227:227*1320=299640,too big;227*1319=299413,rem210. No.\n229:229*1308=299532,rem91. No.\n233:233*1286=299638,too big;233*1285=299405,rem218. No.\n239:239*1253=299467,rem156. No.\n241:241*1243=299563,rem60. No.\n251:251*1193=299443,rem180. No.\n257:257*1165=299405,rem218. No.\n263:263*1139=299557,rem66. No.\n269:269*1113=299397,rem226. No.\n271:271*1105=299455,rem168. No.\n277:277*1081=299437,rem186. No.\n281:281*1066=299546,rem77. No.\n283:283*1058=299414,rem209. No.\n293:293*1022=299446,rem177. No.\n307:307*976=299632,too big;307*975=299325,rem298. No.\n311:311*963=299493,rem130. No.\n313:313*957=299541,rem82. No.\n317:317*945=299565,rem58. No.\n331:331*905=299555,rem68. No.\n337:337*889=299593,rem30. No.\n347:347*863=299461,rem162. No.\n349:349*858=299442,rem181. No.\n353:353*848=299344,rem279. No.\n359:359*834=299406,rem217. No.\n367:367*816=299472,rem151. No.\n373:373*803=299519,rem104. No.\n379:379*790=299410,rem213. No.\n383:383*782=299506,rem117. No.\n389:389*770=299530,rem93. No.\n397:397*754=299338,rem285. No.\n401:401*747=299547,rem76. No.\n409:409*732=299388,rem235. No.\n419:419*715=299585,rem38. No.\n421:421*711=299331,rem292. No.\n431:431*695=299545,rem78. No.\n433:433*691=299203,rem420\u2192 recompute 433*692=299636,too big; 433*691=299203, rem420<433 OK. No.\n439:439*682=299398,rem225. No.\n443:443*676=299468,rem155. No.\n449:449*667=299483,rem140. No.\n457:457*655=299335,rem288. No.\n461:461*649=299189,rem434\u2192 461*650=299650 too big; 461*649=299189, rem434<461. No.\n463:463*647=299561,rem62. No.\n467:467*641=299347,rem276. No.\n479:479*625=299375,rem248. No.\n487:487*615=299505,rem118. No.\n491:491*610=299510,rem113. No.\n499:499*600=299400,rem223. No.\n503:503*595=299285,rem338. No.\n509:509*588=299292,rem331. No.\n521:521*575=299575,rem48. No.\n523:523*572=299156,rem467. No.\n541:541*553=299173,rem450. No.\n547:547*547=299209,rem414. No.\n\nSo 299623 is prime.\n\nn+4=299627. Check.\n299627:\n7:7*42803=299621,rem6. No.\n11:digits2,9,9,6,2,7 alt:7-2+6-9+9-2=9. No.\n13:13*23048=299624,rem3. No.\n17:17*17625=299625,rem2. No.\n19:19*15770=299630,too big;19*15769=299611,rem16. No.\n23:23*13027=299621,rem6. No.\n29:29*10332=299628,too big;29*10331=299599,rem28. No.\n31:31*9665=299615,rem12. No.\n37:37*8098=299626,rem1. No.\n41:41*7308=299628,too big;41*7307=299587,rem40. No.\n43:43*6968=299624,rem3. No.\n47:47*6374=299578,rem49\u219249>47,recompute 47*6375=299625,rem2. No.\n53:53*5653=299609,rem18. No.\n59:59*5078=299602,rem25. No.\n61:61*4912=299632,too big;61*4911=299571,rem56. No.\n67:67*4472=299624,rem3. No.\n71:71*4220=299620,rem7. No.\n73:73*4104=299592,rem35. No.\n79:79*3792=299568,rem59. No.\n83:83*3610=299630,too big;83*3609=299547,rem80. No.\n89:89*3366=299574,rem53. No.\n97:97*3088=299536,rem91. No.\n101:101*2966=299566,rem61. No.\n103:103*2909=299627? 103*2909=103*2900=298700,103*9=927,total299627. Yes! Divisible by103. Not prime.\n\nFails.\n\nNext:299617(\u22617 mod15). n=299617,n+4=299621,n+6=299623.\n299623 prime. Need 299617,299621 prime.\n299617:\n7:7*42802=299614,rem3. No.\n11:digits2,9,9,6,1,7 alt:7-1+6-9+9-2=10. No.\n13:13*23047=299611,rem6. No.\n17:17*17624=299608,rem9. No.\n19:19*15769=299611,rem6. No.\n23:23*13026=299598,rem19. No.\n29:29*10331=299599,rem18. No.\n31:31*9665=299615,rem2. No.\n37:37*8097=299589,rem28. No.\n41:41*7307=299587,rem30. No.\n43:43*6967=299581,rem36. No.\n47:47*6374=299578,rem39. No.\n53:53*5653=299609,rem8. No.\n59:59*5078=299602,rem15. No.\n61:61*4911=299571,rem46. No.\n67:67*4471=299557,rem60. No.\n71:71*4219=299549,rem68. No.\n73:73*4104=299592,rem25. No.\n79:79*3792=299568,rem49. No.\n83:83*3609=299547,rem70. No.\n89:89*3366=299574,rem43. No.\n97:97*3088=299536,rem81. No.\n101:101*2966=299566,rem51. No.\n103:103*2908=299524,rem93. No.\n107:107*2800=299600,rem17. No.\n109:109*2748=299532,rem85. No.\n113:113*2651=299563,rem54. No.\n127:127*2359=299593,rem24. No.\n131:131*2287=299597,rem20. No.\n137:137*2187=299619,too big;137*2186=299482,rem135. No.\n139:139*2155=299545,rem72. No.\n149:149*2010=299490,rem127. No.\n151:151*1984=299584,rem33. No.\n157:157*1908=299556,rem61. No.\n163:163*1838=299594,rem23. No.\n167:167*1794=299598,rem19. No.\n173:173*1731=299463,rem154. No.\n179:179*1673=299467,rem150. No.\n181:181*1655=299555,rem62. No.\n191:191*1568=299488,rem129. No.\n193:193*1552=299536,rem81. No.\n197:197*1520=299440,rem177. No.\n199:199*1505=299495,rem122. No.\n211:211*1420=299620,too big;211*1419=299409,rem208. No.\n223:223*1343=299489,rem128. No.\n227:227*1319=299413,rem204. No.\n229:229*1308=299532,rem85. No.\n233:233*1285=299405,rem212. No.\n239:239*1253=299467,rem150. No.\n241:241*1243=299563,rem54. No.\n251:251*1193=299443,rem174. No.\n257:257*1165=299405,rem212. No.\n263:263*1139=299557,rem60. No.\n269:269*1113=299397,rem220. No.\n271:271*1105=299455,rem162. No.\n277:277*1081=299437,rem180. No.\n281:281*1066=299546,rem71. No.\n283:283*1058=299414,rem203. No.\n293:293*1022=299446,rem171. No.\n307:307*975=299325,rem292. No.\n311:311*963=299493,rem124. No.\n313:313*957=299541,rem76. No.\n317:317*945=299565,rem52. No.\n331:331*905=299555,rem62. No.\n337:337*889=299593,rem24. No.\n347:347*863=299461,rem156. No.\n349:349*858=299442,rem175. No.\n353:353*848=299344,rem273. No.\n359:359*834=299406,rem211. No.\n367:367*816=299472,rem145. No.\n373:373*803=299519,rem98. No.\n379:379*790=299410,rem207. No.\n383:383*782=299506,rem111. No.\n389:389*770=299530,rem87. No.\n397:397*754=299338,rem279. No.\n401:401*747=299547,rem70. No.\n409:409*732=299388,rem229. No.\n419:419*715=299585,rem32. No.\n421:421*711=299331,rem286. No.\n431:431*695=299545,rem72. No.\n433:433*691=299203,rem414. No.\n439:439*682=299398,rem219. No.\n443:443*676=299468,rem149. No.\n449:449*667=299483,rem134. No.\n457:457*655=299335,rem282. No.\n461:461*649=299189,rem428. No.\n463:463*647=299561,rem56. No.\n467:467*641=299347,rem270. No.\n479:479*625=299375,rem242. No.\n487:487*615=299505,rem112. No.\n491:491*610=299510,rem107. No.\n499:499*600=299400,rem217. No.\n503:503*595=299285,rem332. No.\n509:509*588=299292,rem325. No.\n521:521*575=299575,rem42. No.\n523:523*572=299156,rem461. No.\n541:541*553=299173,rem444. No.\n547:547*547=299209,rem408. No.\n\nSo 299617 is prime.\n\nn+4=299621. Check.\n299621:\n7:7*42803=299621! 7*42803=299621. 7*42800=299600,7*3=21,total299621. Yes! Divisible by7. Not prime.\n\nFails.\n\nHmm. Next:299593(\u226113 mod15). n=299593,n+4=299597,n+6=299599.\nThis is getting long. Let me reconsider - maybe I'm missing something. Let me reconsider whether the intended B is really this large.\n\nActually let me continue. But this manual checking is extremely tedious and error-prone. Let me reconsider the problem. Perhaps there's a smaller known answer, or the intended answer just requires the count and largest value which we should get from known data.\n\nGiven the enormous effort, let me reconsider. The number of prime triplets (p,p+4,p+6) below 300000: From OEIS A022004, the count below 10^5 might be known. Actually density of prime triplets. \n\nLet me reconsider. Perhaps I should trust computations that this is genuinely a computational problem. Let me at least find B by continuing, and estimate A.\n\nActually, I realize I should just find the largest triplet more efficiently. Let me continue checking candidates but be smart: I need all three prime. Let me focus.\n\nLet me check 299593 triple:\nn=299593,n+4=299597,n+6=299599.\n299599:check quickly for small factors.\n299599:7:7*42799=299593,rem6.No. 11:digits2,9,9,5,9,9 alt:9-9+5-9+9-2=3.No. 13:13*23046=299598,rem1.No. 17:17*17623=299591,rem8.No. 19:19*15768=299592,rem7.No. 23:23*13026=299598,rem1.No. 29:29*10331=299599!29*10331=290000+9659=299659? wait 29*331=9599, 29*10331=29*10000+29*331=290000+9599=299599. Yes! Divisible by29. So 299599 composite. Fails.\n\nn+6 composite \u2192 fails. \n\nNext 299587(\u22617). n=299587,n+4=299591,n+6=299593.\nNeed 299593 prime first (n+6). Let me check 299593.\n299593:7:7*42799=299593!7*42799=299593. 7*42800=299600,minus7=299593. Yes divisible by7. Composite.\nFails.\n\nNext 299563(\u226113). n=299563,n+4=299567,n+6=299569.\n299563:7:7*42794=299558,rem5.No.11:digits2,9,9,5,6,3 alt:3-6+5-9+9-2=0.Divisible by11! Composite.\nFails.\n\nNext 299557(\u22617). n=299557,n+4=299561,n+6=299563.\n299563 composite. Fails.\n\nNext 299533(\u226113). n=299533,n+4=299537,n+6=299539.\n299533:7:7*42790=299530,rem3.No.11:digits2,9,9,5,3,3 alt:3-3+5-9+9-2=3.No.13:13*23041=299533?13*23041=299533.13*23000=299000,13*41=533,total299533.Yes divisible by13. Composite.\nFails.\n\nNext 299527(\u22617). n=299527,n+4=299531,n+6=299533.\n299533 composite. Fails.\n\nNext 299503(\u226113). n=299503,n+4=299507,n+6=299509.\n299503:7:7*42786=299502,rem1.No.11:digits2,9,9,5,0,3 alt:3-0+5-9+9-2=6.No.13:13*23038=299494,rem9.No.17:17*17618=299506,too big;17*17617=299489,rem14.No.19:19*15763=299497,rem6.No.23:23*13022=299506,too big;23*13021=299483,rem20.No.29:29*10327=299483,rem20.No.31:31*9661=299491,rem12.No.37:37*8094=299478,rem25.No.41:41*7305=299505,too big;41*7304=299464,rem39.No.43:43*6965=299495,rem8.No.47:47*6372=299484,rem19.No.53:53*5651=299503?53*5651=53*5650=299450+53=299503.Yes! Divisible by53. Composite.\nFails.\n\nNext 299497(\u22617). n=299497,n+4=299501,n+6=299503.\n299503 composite. Fails.\n\nNext 299473(\u226113). n=299473,n+4=299477,n+6=299479.\n299473:7:7*42781=299467,rem6.No.11:digits2,9,9,4,7,3 alt:3-7+4-9+9-2=-2.No.13:13*23036=299468,rem5.No.17:17*17616=299472,rem1.No.19:19*15761=299459,rem14.No.23:23*13020=299460,rem13.No.29:29*10326=299454,rem19.No.31:31*9660=299460,rem13.No.37:37*8093=299441,rem32.No.41:41*7304=299464,rem9.No.43:43*6964=299452,rem21.No.47:47*6371=299437,rem36.No.53:53*5650=299450,rem23.No.59:59*5075=299425,rem48.No.61:61*4909=299449,rem24.No.67:67*4469=299423,rem50.No.71:71*4218=299478,too big;71*4217=299407,rem66.No.73:73*4102=299446,rem27.No.79:79*3790=299410,rem63.No.83:83*3608=299464,rem9.No.89:89*3364=299396,rem77.No.97:97*3087=299439,rem34.No.101:101*2965=299465,rem8.No.103:103*2907=299421,rem52.No.107:107*2799=299493,too big;107*2798=299386,rem87.No.109:109*2747=299423,rem50.No.113:113*2650=299450,rem23.No.127:127*2358=299466,rem7.No.131:131*2286=299466,rem7.No.137:137*2186=299482,too big;137*2185=299345,rem128.No.139:139*2154=299406,rem67.No.149:149*2010=299490,too big;149*2009=299341,rem132.No.151:151*1983=299433,rem40.No.157:157*1907=299399,rem74.No.163:163*1837=299431,rem42.No.167:167*1793=299431,rem42.No.173:173*1731=299463,rem10.No.179:179*1673=299467,rem6.No.181:181*1654=299374,rem99.No.191:191*1568=299488,too big;191*1567=299297,rem176.No.193:193*1551=299343,rem130.No.197:197*1520=299440,rem33.No.199:199*1505=299495,too big;199*1504=299296,rem177.No.211:211*1419=299409,rem64.No.223:223*1343=299489,too big;223*1342=299266,rem207.No.227:227*1319=299413,rem60.No.229:229*1307=299303,rem170.No.233:233*1285=299405,rem68.No.239:239*1253=299467,rem6.No.241:241*1242=299322,rem151.No.251:251*1193=299443,rem30.No.257:257*1165=299405,rem68.No.263:263*1138=299294,rem179.No.269:269*1113=299397,rem76.No.271:271*1105=299455,rem18.No.277:277*1081=299437,rem36.No.281:281*1065=299265,rem208.No.283:283*1058=299414,rem59.No.293:293*1022=299446,rem27.No.307:307*975=299325,rem148.No.311:311*963=299493,too big;311*962=299182,rem291.No.313:313*956=299228,rem245.No.317:317*944=299248,rem225.No.331:331*904=299224,rem249.No.337:337*888=299256,rem217.No.347:347*863=299461,rem12.No.349:349*858=299442,rem31.No.353:353*848=299344,rem129.No.359:359*834=299406,rem67.No.367:367*816=299472,rem1.No.373:373*803=299519,too big;373*802=299146,rem327.No.379:379*790=299410,rem63.No.383:383*782=299506,too big;383*781=299123,rem350.No.389:389*770=299530,too big;389*769=299141,rem332.No.397:397*754=299338,rem135.No.401:401*746=299146,rem327.No.409:409*732=299388,rem85.No.419:419*714=299166,rem307.No.421:421*711=299331,rem142.No.431:431*694=299114,rem359.No.433:433*691=299203,rem270.No.439:439*682=299398,rem75.No.443:443*676=299468,rem5.No.449:449*667=299483,too big;449*666=299034,rem439.No.457:457*655=299335,rem138.No.461:461*649=299189,rem284.No.463:463*646=299098,rem375.No.467:467*641=299347,rem126.No.479:479*625=299375,rem98.No.487:487*614=299018,rem455.No.491:491*610=299510,too big;491*609=299019,rem454.No.499:499*600=299400,rem73.No.503:503*595=299285,rem188.No.509:509*588=299292,rem181.No.521:521*574=299054,rem419.No.523:523*572=299156,rem317.No.541:541*553=299173,rem300.No.547:547*547=299209,rem264.No.\n\nSo 299473 is prime.\n\nn+4=299477. Check.\n299477:7:7*42782=299474,rem3.No.11:digits2,9,9,4,7,7 alt:7-7+4-9+9-2=2.No.13:13*23036=299468,rem9.No.17:17*17616=299472,rem5.No.19:19*15761=299459,rem18.No.23:23*13020=299460,rem17.No.29:29*10326=299454,rem23.No.31:31*9660=299460,rem17.No.37:37*8094=299478,too big;37*8093=299441,rem36.No.41:41*7304=299464,rem13.No.43:43*6964=299452,rem25.No.47:47*6372=299484,too big;47*6371=299437,rem40.No.53:53*5650=299450,rem27.No.59:59*5075=299425,rem52.No.61:61*4909=299449,rem28.No.67:67*4469=299423,rem54.No.71:71*4218=299478,too big;71*4217=299407,rem70.No.73:73*4102=299446,rem31.No.79:79*3790=299410,rem67.No.83:83*3608=299464,rem13.No.89:89*3364=299396,rem81.No.97:97*3087=299439,rem38.No.101:101*2965=299465,rem12.No.103:103*2907=299421,rem56.No.107:107*2798=299386,rem91.No.109:109*2747=299423,rem54.No.113:113*2650=299450,rem27.No.127:127*2358=299466,rem11.No.131:131*2286=299466,rem11.No.137:137*2185=299345,rem132.No.139:139*2154=299406,rem71.No.149:149*2009=299341,rem136.No.151:151*1983=299433,rem44.No.157:157*1907=299399,rem78.No.163:163*1837=299431,rem46.No.167:167*1793=299431,rem46.No.173:173*1731=299463,rem14.No.179:179*1673=299467,rem10.No.181:181*1654=299374,rem103.No.191:191*1567=299297,rem180.No.193:193*1551=299343,rem134.No.197:197*1520=299440,rem37.No.199:199*1504=299296,rem181.No.211:211*1419=299409,rem68.No.223:223*1342=299266,rem211.No.227:227*1319=299413,rem64.No.229:229*1307=299303,rem174.No.233:233*1285=299405,rem72.No.239:239*1253=299467,rem10.No.241:241*1242=299322,rem155.No.251:251*1193=299443,rem34.No.257:257*1165=299405,rem72.No.263:263*1138=299294,rem183.No.269:269*1113=299397,rem80.No.271:271*1105=299455,rem22.No.277:277*1081=299437,rem40.No.281:281*1065=299265,rem212.No.283:283*1058=299414,rem63.No.293:293*1022=299446,rem31.No.307:307*975=299325,rem152.No.311:311*962=299182,rem295.No.313:313*956=299228,rem249.No.317:317*944=299248,rem229.No.331:331*904=299224,rem253.No.337:337*888=299256,rem221.No.347:347*863=299461,rem16.No.349:349*858=299442,rem35.No.353:353*848=299344,rem133.No.359:359*834=299406,rem71.No.367:367*816=299472,rem5.No.373:373*802=299146,rem331.No.379:379*790=299410,rem67.No.383:383*781=299123,rem354.No.389:389*769=299141,rem336.No.397:397*754=299338,rem139.No.401:401*746=299146,rem331.No.409:409*732=299388,rem89.No.419:419*714=299166,rem311.No.421:421*711=299331,rem146.No.431:431*694=299114,rem363.No.433:433*691=299203,rem274.No.439:439*682=299398,rem79.No.443:443*676=299468,rem9.No.449:449*666=299034,rem443.No.457:457*655=299335,rem142.No.461:461*649=299189,rem288.No.463:463*646=299098,rem379.No.467:467*641=299347,rem130.No.479:479*625=299375,rem102.No.487:487*614=299018,rem459.No.491:491*609=299019,rem458.No.499:499*600=299400,rem77.No.503:503*595=299285,rem192.No.509:509*588=299292,rem185.No.521:521*574=299054,rem423.No.523:523*572=299156,rem321.No.541:541*553=299173,rem304.No.547:547*547=299209,rem268.No.\n\nSo 299477 is prime.\n\nn+6=299479. Check.\n299479:7:7*42782=299474,rem5.No.11:digits2,9,9,4,7,9 alt:9-7+4-9+9-2=4.No.13:13*23036=299468,rem11.No.17:17*17616=299472,rem7.No.19:19*15762=299478,rem1.No.23:23*13020=299460,rem19.No.29:29*10327=299483,too big;29*10326=299454,rem25.No.31:31*9660=299460,rem19.No.37:37*8093=299441,rem38.No.41:41*7304=299464,rem15.No.43:43*6964=299452,rem27.No.47:47*6371=299437,rem42.No.53:53*5650=299450,rem29.No.59:59*5075=299425,rem54.No.61:61*4909=299449,rem30.No.67:67*4469=299423,rem56.No.71:71*4217=299407,rem72\u219272>71,recompute71*4218=299478,rem1.No.73:73*4102=299446,rem33.No.79:79*3790=299410,rem69.No.83:83*3608=299464,rem15.No.89:89*3364=299396,rem83.No.97:97*3087=299439,rem40.No.101:101*2965=299465,rem14.No.103:103*2907=299421,rem58.No.107:107*2798=299386,rem93.No.109:109*2747=299423,rem56.No.113:113*2650=299450,rem29.No.127:127*2358=299466,rem13.No.131:131*2286=299466,rem13.No.137:137*2185=299345,rem134.No.139:139*2154=299406,rem73.No.149:149*2009=299341,rem138.No.151:151*1983=299433,rem46.No.157:157*1907=299399,rem80.No.163:163*1837=299431,rem48.No.167:167*1793=299431,rem48.No.173:173*1731=299463,rem16.No.179:179*1673=299467,rem12.No.181:181*1654=299374,rem105.No.191:191*1567=299297,rem182.No.193:193*1551=299343,rem136.No.197:197*1520=299440,rem39.No.199:199*1504=299296,rem183.No.211:211*1419=299409,rem70.No.223:223*1342=299266,rem213.No.227:227*1319=299413,rem66.No.229:229*1307=299303,rem176.No.233:233*1285=299405,rem74.No.239:239*1253=299467,rem12.No.241:241*1242=299322,rem157.No.251:251*1193=299443,rem36.No.257:257*1165=299405,rem74.No.263:263*1138=299294,rem185.No.269:269*1113=299397,rem82.No.271:271*1105=299455,rem24.No.277:277*1081=299437,rem42.No.281:281*1065=299265,rem214.No.283:283*1058=299414,rem65.No.293:293*1022=299446,rem33.No.307:307*975=299325,rem154.No.311:311*962=299182,rem297.No.313:313*956=299228,rem251.No.317:317*944=299248,rem231.No.331:331*904=299224,rem255.No.337:337*888=299256,rem223.No.347:347*863=299461,rem18.No.349:349*858=299442,rem37.No.353:353*848=299344,rem135.No.359:359*834=299406,rem73.No.367:367*816=299472,rem7.No.373:373*802=299146,rem333.No.379:379*790=299410,rem69.No.383:383*781=299123,rem356.No.389:389*769=299141,rem338.No.397:397*754=299338,rem141.No.401:401*746=299146,rem333.No.409:409*732=299388,rem91.No.419:419*714=299166,rem313.No.421:421*711=299331,rem148.No.431:431*694=299114,rem365.No.433:433*691=299203,rem276.No.439:439*682=299398,rem81.No.443:443*676=299468,rem11.No.449:449*666=299034,rem445.No.457:457*655=299335,rem144.No.461:461*649=299189,rem290.No.463:463*646=299098,rem381.No.467:467*641=299347,rem132.No.479:479*625=299375,rem104.No.487:487*614=299018,rem461.No.491:491*609=299019,rem460.No.499:499*600=299400,rem79.No.503:503*595=299285,rem194.No.509:509*588=299292,rem187.No.521:521*574=299054,rem425.No.523:523*572=299156,rem323.No.541:541*553=299173,rem306.No.547:547*547=299209,rem270.No.\n\nSo 299479 is prime!\n\nSo n=299473: 299473, 299477, 299479 all prime! This is a triplet!\n\nWait, but we need (n, n+4, n+6). n=299473, n+4=299477, n+6=299479. Yes all prime! Great, so B could be 299473.\n\nBut wait, I need to double-check we didn't skip a larger valid one. We checked all candidates from 299983 down to 299473 and found none valid until 299473. Let me make sure I covered all n\u22617 or 13 mod15 odd values in (299473, 300000).\n\nThe candidates I checked descending:\n299983(13),299977(7),299953(13),299947(7),299923(13),299917(7),299893(13),299887(7),299863(13),299857(7),299833(13),299827(7),299803(13),299797(7),299773(13),299767(7),299743(13),299737(7),299713(13),299707(7),299683(13),299677(7),299653(13),299647(7),299623(13),299617(7),299593(13),299587(7),299563(13),299557(7),299533(13),299527(7),299503(13),299497(7),299473(13).\n\nWait, I need to double check the mod15 sequence spacing. Values \u226113 mod15 odd and \u22617 mod15 odd. Between consecutive, differences. Let me verify I haven't skipped any.\n\nNumbers \u22617 mod15: ...,299947,299962,299977,299992. Odd ones \u22617 mod15: 299947,299977 (since 299962 even, 299992 even). Wait 299947 and 299977 differ by 30. And 299917 (299917=299947-30). Yes step 30 for odd within same residue class mod15? Actually mod30 the odd residues. n\u22617 mod15 and odd \u2192 n\u22617 mod30 (7 is odd) \u2192 7,37,...,299917,299947,299977. Yes step 30.\n\nn\u226113 mod15 and odd \u2192 n\u226113 mod30 \u219213,43,...,299953,299983... and 299923,299893,...,299473,299503... Let me check: 299473 mod30: 299473=30*9982+13, yes \u226113 mod30. Good.\n\nSo candidates \u22617 mod30: ...,299467,299497,299527,299557,299587,299617,299647,299677,299707,299737,299767,299797,299827,299857,299887,299917,299947,299977. \nCandidates \u226113 mod30: ...,299473,299503,299533,299563,299593,299623,299653,299683,299713,299743,299773,299803,299833,299863,299893,299923,299953,299983.\n\nBetween 299473 and 300000, \u226113 mod30: 299473,299503,...,299983. \u22617 mod30: 299497,...,299977.\n\nI checked all of these. Good. And 299473 is the winner as the largest triplet.\n\nActually wait, I should double check there's nothing between 299983 and 300000. Next \u22617 mod30 after 299977 is 300007 (>300000). Next \u226113 mod30 after 299983 is 300013(>300000). So 299983 was the top candidate. Good.\n\nSo B = 299473.\n\nNow I need A, the count of all such n in [2,300000]. This requires enumerating all prime triplets (p,p+4,p+6) up to 300000. That's a large count \u2014 I cannot reliably compute by hand.\n\nHmm. This is the hard part. Let me recall the OEIS sequence A022004 and count terms \u2264 300000.\n\nLet me recall/reconstruct A022004 more fully. Actually, I recall the count of such triplets. Let me think about the density. The number of prime triplets of a given admissible pattern below N is ~ C * N/(ln N)^3. For N=300000, ln N \u2248 12.6, (ln N)^3 \u2248 2000. The constant for triplets (Hardy-Littlewood) is about 2.858 (the triplet constant). So estimate ~ 2.858 * 300000/2000 \u2248 2.858*150 \u2248 429? That seems too high. Wait, the constant for triplet type is around 2.858... but this includes the factor.\n\nActually the Hardy-Littlewood prediction for number of prime triplets (p, p+4, p+6) below x is approximately (9/2)\u00b7C_3\u00b7\u222b... hmm this is getting complicated. Let me instead recall actual known counts.\n\nActually I recall that the number of prime triplets (both patterns combined) below 10^6 is around 1600+. For a single pattern below 300000, maybe around 250-280.\n\nI genuinely can't produce the exact count reliably by hand. Given the constraints, let me try to recall the actual value.\n\nHmm, given the difficulty, I suspect the intended answer wants us to actually count. Let me reconsider - actually the number of terms in A022004 up to 300000.\n\nLet me try to recall data. The number of prime triplets (p,p+4,p+6) with p < 10^5. I think there are around 100 or so. Let me estimate better.\n\nHL constant for this specific triplet: The number of triplets \u2248 2\u00b7C \u00b7 x/(ln x)^3 where the specific constant... For the pattern (0,4,6), the singular series. Admissible: residues mod 2: {0,4,6} all even offsets \u2192 all odd, fine. Mod 3: {0,4,6}\u2261{0,1,0} \u2192 covers residues 0 and 1, misses 2. Good (admissible). \n\nThe count function \u03c0_3(x) for triplets. Known: number of prime triplets (of form p,p+2,p+6 or p,p+4,p+6) below 10^6 is 1444 total? \n\nLet me just try to actually estimate via the HL constant. For pattern (0,4,6):\nConstant = product over primes... The Hardy-Littlewood conjecture: \nN(x) ~ H \u00b7 \u222b_2^x dt/(ln t)^3\nwhere H = 2^{... } \u220f_p (1 - \u03bd_p/p)/(1-1/p)^3, \u03bd_p = number of residues covered.\n\nFor p=2: offsets {0,4,6}\u2261{0,0,0} mod2, so \u03bd_2=1. Factor: (1-1/2)/(1-1/2)^3 = (1/2)/(1/8)=4.\nFor p=3: \u03bd_3=2 (residues 0,1). Factor: (1-2/3)/(1-1/3)^3 = (1/3)/(8/27)=(1/3)(27/8)=9/8=1.125.\nFor p\u22655: offsets {0,4,6} distinct mod p (as long as p doesn't divide differences: differences are 4,6,2; for p=5, {0,4,6}\u2261{0,4,1} distinct, \u03bd=3; p\u22655 generally \u03bd_p=3). Factor: (1-3/p)/(1-1/p)^3.\n\nSo H = 4 \u00b7 1.125 \u00b7 \u220f_{p\u22655} (1-3/p)/(1-1/p)^3 = 4.5 \u00b7 \u220f_{p\u22655}(1-3/p)/(1-1/p)^3.\n\nThe product \u220f_{p\u22655}(1-3/p)/(1-1/p)^3 \u2248 let me compute the known triplet constant. The Hardy-Littlewood constant for triplets is often quoted. \u220f_{p\u22655}(1-3/p)/(1-1/p)^3. Let me compute a few terms:\np=5: (1-0.6)/(0.8)^3=0.4/0.512=0.78125.\np=7:(1-3/7)/(6/7)^3=(4/7)/(216/343)=(0.5714)/(0.6297)=0.9075.\np=11:(1-3/11)/(10/11)^3=(8/11)/(1000/1331)=(0.7273)/(0.7513)=0.9681.\np=13:(10/13)/(12/13)^3=(0.7692)/(0.7865)=0.9780.\np=17:(14/17)/(16/17)^3=(0.8235)/(0.8340)=0.9874.\np=19:(16/19)/(18/19)^3=(0.8421)/(0.8501)=0.9906.\np=23:(20/23)/(22/23)^3=(0.8696)/(0.8747)=0.9942.\nContinuing, product converges. Let me multiply:\n0.78125*0.9075=0.7090.\n*0.9681=0.6864.\n*0.9780=0.6713.\n*0.9874=0.6628.\n*0.9906=0.6566.\n*0.9942=0.6528.\nRemaining primes contribute factors approaching 1. Roughly product from 29 onward: each factor \u22481-3/p\u00b2... actually (1-3/p)/(1-1/p)^3 \u2248 1 - 3/p\u00b2+... for large p. Sum of 3/p\u00b2 for p\u226529 is small \u2248 3*(0.001+...) \u2248 0.004. So remaining product \u2248 0.996. So total \u2248 0.6528*0.996\u22480.650.\n\nSo H \u2248 4.5 * 0.650 \u2248 2.925.\n\nNow N(x) \u2248 H \u00b7 \u222b_2^x dt/(ln t)^3.\nFor x=300000. \u222b_2^x dt/(ln t)^3 \u2248 x/(ln x)^3 \u00b7 (1 + 3/ln x + ...) roughly. Actually let me just use Li_3 approx. ln(300000)=12.612. (ln x)^3=2005. x/(ln x)^3=300000/2005=149.6. With correction factor (1+3/lnx+12/(lnx)^2)\u22481+0.238+0.0755\u22481.31. So integral\u2248149.6*1.31\u2248196. Hmm but the integral from 2 accumulates; let me be more careful. Actually the asymptotic \u222b_2^x dt/(ln t)^3 \u2248 x/(ln x)^3 [1 + 3/ln x + 12/(ln x)^2 + ...]. \n\nSo N(x) \u2248 2.925 * 196 \u2248 573?? That seems too high.\n\nWait, I think I'm double counting the constant. Let me reconsider. The HL conjecture counts ordered by p, giving number of p \u2264 x. The formula is N(x) ~ H \u222b_2^x dt/(ln t)^3 where H is the singular series as I defined. Hmm but 573 seems too many.\n\nLet me sanity check against known: number of prime triplets (p,p+4,p+6) below 10^5. Let me count from the beginning of A022004 that I listed:\n7,13,37,67,97,103,193,223,277,307,457,613,823,853,877,1087,1297,1423,1483,1663,1693,1783,1867,1993,2083,2137,2377,2683,2707,2797,3163,3253,3457,3463,3847,4153,4513,4783,5227,5413,5437,5647,5653,5827,6547,6553,6653,6823,7057,7873,8287,8353,8677,...\n\nThat's already ~54 terms below 8677. Extrapolating density, below 10^5 there'd be maybe... the density decreases. Rough: by 8677 we have 54. The count grows roughly like the integral. Actually if there are 54 below ~8700, below 300000 (35x larger, but log^3 grows) ... \n\nHmm, my HL estimate of ~570 for 300000 might be reasonable-ish. But I really can't count exactly by hand.\n\nThis is a serious problem. Given the format expects an exact integer, and I've determined B=299473, I need A.\n\nGiven I cannot reliably enumerate hundreds of triplets by hand, let me reconsider whether the answer is meant to be computed and I should provide my best estimate.\n\nLet me reconsider the HL estimate more carefully, because the exact count matters a lot for A+B.\n\nActually, let me reconsider. The integral \u222b_2^x dt/(ln t)^3. Let me compute more carefully numerically. Alternatively known values: the actual count of these triplets.\n\nI found reference in memory: The number of prime triplets of the form (p,p+4,p+6) below 10^6 is around... Actually, let me use the combined data. Total prime triplets below 10^6 (both forms) \u2248 4055? No...\n\nI'll go with my HL estimate but refine. Actually let me reconsider whether H\u22482.925 is right. The commonly cited \"triplet constant\" for these is about 2.858. Let me use ~2.86.\n\nNow the integral. Let me compute \u222b_2^{300000} dt/(ln t)^3 numerically via segments.\n\nActually, let me instead directly estimate the count differently \u2014 count = H * average of 1/(ln p)^3 style. \n\nLet me just numerically integrate \u222b dt/(ln t)^3 from say 10 to 300000.\nUse substitution. Let me evaluate integrand at several points and use the fact.\n\nValues of 1/(ln t)^3:\nt=10: ln=2.303, cube=12.2, 1/=0.0820\nt=100: ln=4.605,cube=97.6,1/=0.01024\nt=1000: ln=6.908,cube=329.6,1/=0.003034\nt=10000: ln=9.210,cube=781.2,1/=0.001280\nt=100000: ln=11.513,cube=1526,1/=0.000655\nt=300000: ln=12.612,cube=2006,1/=0.000499\n\nThis integrand. Let me integrate piecewise (log-spaced is tricky). Let me do linear segments with trapezoid on finer grid.\n\nLet me use grid: t=2,10,100,1000,10000,100000,300000 \u2014 too coarse because integrand changes and t-range huge.\n\nBetter: the dominant contribution is at large t (since dt is large there). Let me integrate from 10^4 to 3*10^5 more carefully, and add earlier part.\n\nFrom 10^4 to 10^5: integrand goes 0.00128 to 0.000655. This is over an interval that's mostly weighted toward larger t. Let me subdivide:\nt=10000:0.001280\nt=20000: ln=9.903,cube=971.2,1/=0.001030\nt=30000: ln=10.309,cube=1095.6,1/=0.000913\nt=50000: ln=10.820,cube=1266.9,1/=0.000789\nt=70000: ln=11.156,cube=1388.5,1/=0.000720\nt=100000:0.000655\nt=150000: ln=11.918,cube=1693,1/=0.000591\nt=200000: ln=12.206,cube=1819,1/=0.000550\nt=300000:0.000499\n\nIntegrate 10000\u219220000: avg~(0.00128+0.00103)/2=0.001155 *10000=11.55\n20000\u219230000: avg(0.00103+0.000913)/2=0.000972*10000=9.72\n30000\u219250000: avg(0.000913+0.000789)/2=0.000851*20000=17.02\n50000\u219270000: avg(0.000789+0.000720)/2=0.0007545*20000=15.09\n70000\u2192100000: avg(0.000720+0.000655)/2=0.0006875*30000=20.63\n100000\u2192150000: avg(0.000655+0.000591)/2=0.000623*50000=31.15\n150000\u2192200000: avg(0.000591+0.000550)/2=0.0005705*50000=28.53\n200000\u2192300000: avg(0.000550+0.000499)/2=0.0005245*100000=52.45\nSum 10000\u2192300000: 11.55+9.72+17.02+15.09+20.63+31.15+28.53+52.45=186.14\n\nNow 1000\u219210000:\nt=1000:0.003034\nt=2000:ln=7.601,cube=439.2,1/=0.002277\nt=3000:ln=8.006,cube=513.2,1/=0.001949\nt=5000:ln=8.517,cube=617.8,1/=0.001619\nt=7000:ln=8.854,cube=694.0,1/=0.001441\nt=10000:0.001280\n1000\u21922000:avg(0.003034+0.002277)/2=0.002656*1000=2.656\n2000\u21923000:avg(0.002277+0.001949)/2=0.002113*1000=2.113\n3000\u21925000:avg(0.001949+0.001619)/2=0.001784*2000=3.568\n5000\u21927000:avg(0.001619+0.001441)/2=0.001530*2000=3.060\n7000\u219210000:avg(0.001441+0.001280)/2=0.0013605*3000=4.082\nSum:2.656+2.113+3.568+3.060+4.082=15.48\n\n100\u21921000:\nt=100:0.01024\nt=200:ln=5.298,cube=148.7,1/=0.006724\nt=300:ln=5.704,cube=185.6,1/=0.005388\nt=500:ln=6.215,cube=240.0,1/=0.004167\nt=700:ln=6.551,cube=281.2,1/=0.003557\nt=1000:0.003034\n100\u2192200:avg(0.01024+0.006724)/2=0.008482*100=0.848\n200\u2192300:avg(0.006724+0.005388)/2=0.006056*100=0.606\n300\u2192500:avg(0.005388+0.004167)/2=0.004778*200=0.956\n500\u2192700:avg(0.004167+0.003557)/2=0.003862*200=0.772\n700\u21921000:avg(0.003557+0.003034)/2=0.003296*300=0.989\nSum:0.848+0.606+0.956+0.772+0.989=4.17\n\n10\u2192100:\nt=10:0.0820\nt=20:ln=2.996,cube=26.9,1/=0.03717\nt=30:ln=3.401,cube=39.3,1/=0.02543\nt=50:ln=3.912,cube=59.9,1/=0.01670\nt=70:ln=4.248,cube=76.7,1/=0.01304\nt=100:0.01024\n10\u219220:avg(0.0820+0.03717)/2=0.05959*10=0.596\n20\u219230:avg(0.03717+0.02543)/2=0.03130*10=0.313\n30\u219250:avg(0.02543+0.01670)/2=0.02107*20=0.421\n50\u219270:avg(0.01670+0.01304)/2=0.01487*20=0.297\n70\u2192100:avg(0.01304+0.01024)/2=0.01164*30=0.349\nSum:0.596+0.313+0.421+0.297+0.349=1.976\n\nTotal integral 10\u2192300000 \u2248 186.14+15.48+4.17+1.976=207.77. Plus small from 2\u219210 (~0.5). \u2248 208.\n\nTrapezoid overestimates for convex decreasing function, so maybe ~200.\n\nThen N \u2248 H * integral \u2248 2.86 * 200 \u2248 572. Hmm.\n\nBut wait, this HL formula might overestimate for finite ranges, and my constant H might be off. Also trapezoidal overestimate. Actual count likely somewhat less, maybe ~500-560.\n\nI cannot pin this to an exact integer. This is problematic.\n\nGiven the situation, I'll make my best estimate. Actually, let me reconsider\u2014maybe actual known count. Let me reconsider: I found 54 terms below 8677 in my recalled list, but I'm not confident that list is complete/correct.\n\nGiven uncertainty, and that this problem clearly expects a computed answer, let me provide my best estimate for A. Based on HL with finite-size correction, I'll estimate A \u2248 540.\n\nActually, let me reconsider the known count. I'll try to recall: A022004 has how many terms below 300000? \n\nI genuinely don't know. Let me go with the HL estimate around 530-570. I'll pick A \u2248 540 as a compromise, acknowledging uncertainty. Actually the trapezoid overestimate suggests lower. Let me use integral\u2248200, N\u22482.86*200=572, then reduce for overestimate to ~540.\n\nHmm, but the actual constant: let me reconsider if H should be 2.858. Different sources give the triplet constant differently. Given my computed H\u22482.925, and the true as"
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