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 A139547 Triangle read by rows: T(n,k) = A003418(A010766). 5
 1, 2, 1, 6, 1, 1, 12, 2, 1, 1, 60, 2, 1, 1, 1, 60, 6, 2, 1, 1, 1, 420, 6, 2, 1, 1, 1, 1, 840, 12, 2, 2, 1, 1, 1, 1, 2520, 12, 6, 2, 1, 1, 1, 1, 1, 2520, 60, 6, 2, 2, 1, 1, 1, 1, 1, 27720, 60, 6, 2, 2, 1, 1, 1, 1, 1, 1, 27720, 60, 12, 6, 2, 2, 1, 1, 1, 1, 1, 1, 360360, 60, 12, 6, 2, 2, 1, 1, 1, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS This triangle fits the formula of I. Vardi in the Mathworld link about the von Mangoldt function. That formula is the basis for Chebyshev's estimate for the number of primes. REFERENCES I. Vardi, Computational Recreations in Mathematica. Addison-Wesley, Redwood City, CA, 1991, p. 155. LINKS Eric Weisstein's World of Mathematics, Mangoldt Function.. FORMULA From Mats Granvik, Jun 05 2016: (Start) T(n,k)=A003418(floor(n/k)). Recurrence involving log(n!): Let s=1. T(n, k) = if k = 1 then log(n!) - Sum_{i=2..n} T(n, i)/i^(s - 1) else if n >= k then T(floor(n/k), 1) else 0 else 0. Recurrence involving the Riemann zeta function: Let z = 1. Let a = the series expansion of zeta(s) at z. Let ss -> Infinity. Let s = z + 1/ss. Then T(n,k) is generated by the recurrence: a + Ts(n, k) = if k = 1 then n*zeta(s) - Sum_{i=2..n} Ts(n, i)/i^(s - 1) else if n >= k then Ts(floor(n/k), 1) else 0 else 0. (End) EXAMPLE Triangle begins: 1; 2,1; 6,1,1; 12,2,1,1; 60,2,1,1,1; 60,6,2,1,1,1; 420,6,2,1,1,1,1; 840,12,2,2,1,1,1,1; 2520,12,6,2,1,1,1,1,1; 2520,60,6,2,2,1,1,1,1,1; 27720,60,6,2,2,1,1,1,1,1,1; 27720,60,12,6,2,2,1,1,1,1,1,1; 360360,60,12,6,2,2,1,1,1,1,1,1,1; ... MATHEMATICA nn = 13; a = Exp[Accumulate[MangoldtLambda[Range[nn]]]]; Flatten[Table[Table[a[[Floor[n/k]]], {k, 1, n}], {n, 1, nn}]][[1 ;; 89]] (*As a limit of a recurrence*) Clear[t, s, n, k, z, nn, ss, a, aa]; (*z=1 corresponds to Zeta[1], z=2 corresponds to Zeta[2], z=ZetaZero[1] corresponds to Zeta[ZetaZero[1]], etc.*) z = 1; a = Normal[Series[Zeta[s], {s, z, 0}]]; ss = 10^40; s = N[z + 1/ss, 10^2]; nn = 13; t[n_, k_] := t[n, k] = If[k == 1, n*Zeta[s] - Sum[t[n, i]/i^(s - 1), {i, 2, n}], If[n >= k, t[Floor[n/k], 1], 0], 0]; aa = Table[Table[If[n >= k, t[n, k] - a, 0], {k, 1, n}], {n, 1, nn}]; Flatten[Round[Exp[aa]]][[1 ;; 89]] (* Mats Granvik, Jun 05 2016 *) CROSSREFS Cf. A000142, A010766, A014963, A003418, A139550, A139552, A139554. Sequence in context: A094673 A196839 A089808 * A126342 A229818 A082388 Adjacent sequences:  A139544 A139545 A139546 * A139548 A139549 A139550 KEYWORD nonn,tabl AUTHOR Mats Granvik, Apr 27 2008, May 07 2008 EXTENSIONS Edited by Mats Granvik, Jun 28 2009 Further edits from N. J. A. Sloane, Jul 03 2009 STATUS approved

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