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

Author

N. J. A. Sloane

Keywords

• nonn
• frac

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

Examples

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

Links

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

Crossrefs

Cf. A017676 (denominator), A013664, A013665.

Programs

Magma

[Numerator(DivisorSigma(6,n)/n^6): n in [1..20]]; // G. C. Greubel, Nov 07 2018

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 *)

PARI

vector(20, n, numerator(sigma(n, 6)/n^6)) \\ G. C. Greubel, Nov 07 2018

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)