A017691 Numerator of sum of -14th powers of divisors of n.

1, 16385, 4782970, 268451841, 6103515626, 39184481725, 678223072850, 4398314962945, 22876797237931, 10000610353201, 379749833583242, 213999516991295, 3937376385699290, 5556342524323625, 5838586426737844, 72061992352890881, 168377826559400930, 374836322743499435

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..10000

Crossrefs

Cf. A017692 (denominator), A013672, A013673.

Programs

Magma

[Numerator(DivisorSigma(14,n)/n^14): n in [1..20]]; // G. C. Greubel, Nov 06 2018

Mathematica

Table[Numerator[DivisorSigma[14, n]/n^14], {n, 1, 20}] (* G. C. Greubel, Nov 06 2018 *)

PARI

vector(20, n, numerator(sigma(n, 14)/n^14)) \\ G. C. Greubel, Nov 06 2018

Formula

From Amiram Eldar, Apr 02 2024: (Start)

sup_{n>=1} a(n)/A017692(n) = zeta(14) (A013672).

Dirichlet g.f. of a(n)/A017692(n): zeta(s)*zeta(s+14).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017692(k) = zeta(15) (A013673). (End)