A017709 Numerator of sum of -23rd powers of divisors of n.

1, 8388609, 94143178828, 70368752566273, 11920928955078126, 65810859767097521, 27368747340080916344, 590295880727458217985, 8862938119746644274757, 50000005960464481733367, 895430243255237372246532, 1656184514187480740117011, 41753905413413116367045798

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

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A017710 (denominator).

Programs

Magma

[Numerator(DivisorSigma(23,n)/n^23): n in [1..20]]; // G. C. Greubel, Nov 03 2018

Mathematica

Table[Numerator[Total[Divisors[n]^-23]],{n,12}] (* Harvey P. Dale, Oct 19 2012 *)
Table[Numerator[DivisorSigma[23, n]/n^23], {n, 1, 20}] (* G. C. Greubel, Nov 03 2018 *)

PARI

a(n) = numerator(sigma(n, 23)/n^23); \\ G. C. Greubel, Nov 03 2018

Formula

From Amiram Eldar, Apr 02 2024: (Start)

sup_{n>=1} a(n)/A017710(n) = zeta(23).

Dirichlet g.f. of a(n)/A017710(n): zeta(s)*zeta(s+23).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017710(k) = zeta(24). (End)