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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

Table of n, a(n) for n=0..88.

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: A295315 A089808 A290318 * A323855 A126342 A229818

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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Last modified May 31 19:40 EDT 2020. Contains 334748 sequences. (Running on oeis4.)