A017675 Numerator of sum of -6th powers of divisors of n.

1, 65, 730, 4161, 15626, 23725, 117650, 266305, 532171, 101569, 1771562, 506255, 4826810, 3823625, 2281396, 17043521, 24137570, 34591115, 47045882, 32509893, 85884500, 57575765, 148035890, 97201325, 244156251, 12067025, 387952660, 244770825, 594823322

Offset

1,2

Comments

Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

Formula

Numerators of coefficients in expansion of Sum_{k>=1} x^k/(k^6*(1 - x^k)). - Ilya Gutkovskiy, May 25 2018

From Amiram Eldar, Apr 02 2024: (Start)

sup_{n>=1} a(n)/A017676(n) = zeta(6) (A013664).

Dirichlet g.f. of a(n)/A017676(n): zeta(s)*zeta(s+6).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017676(k) = zeta(7) (A013665). (End)

Example

1, 65/64, 730/729, 4161/4096, 15626/15625, 23725/23328, 117650/117649, 266305/262144, ...

Mathematica

A017675[n_Integer] := Numerator[DivisorSigma[-6, n]]; Table[A017675[n], {n, 100}] (* Vladimir Joseph Stephan Orlovsky, Jul 22 2011 *)
Table[Numerator[DivisorSigma[6, n]/n^6], {n, 1, 20}] (* G. C. Greubel, Nov 07 2018 *)

Prog

(PARI) vector(20, n, numerator(sigma(n, 6)/n^6)) \\ G. C. Greubel, Nov 07 2018
(Magma) [Numerator(DivisorSigma(6, n)/n^6): n in [1..20]]; // G. C. Greubel, Nov 07 2018

Crossrefs

Cf. A017676 (denominator), A013664, A013665.

Sequence in context: A321562 A034680 A351301 * A013954 A294301 A343508

Adjacent sequences: A017672 A017673 A017674 * A017676 A017677 A017678

Keyword

nonn,frac

Author

N. J. A. Sloane

Status

approved