A017705 Numerator of sum of -21st powers of divisors of n.

1, 2097153, 10460353204, 4398048608257, 476837158203126, 609360030634117, 558545864083284008, 9223376434903384065, 109418989141972712413, 500000238418580150139, 7400249944258160101212, 11501285462682212701357, 247064529073450392704414, 146419516812481413403653

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. A017706 (denominator).

Programs

Magma

[Numerator(DivisorSigma(21,n)/n^21): n in [1..20]]; // G. C. Greubel, Nov 05 2018

Mathematica

Table[Numerator[DivisorSigma[21, n]/n^21], {n, 1, 20}] (* G. C. Greubel, Nov 05 2018 *)

PARI

vector(20, n, numerator(sigma(n, 21)/n^21)) \\ G. C. Greubel, Nov 05 2018

Formula

From Amiram Eldar, Apr 02 2024: (Start)

sup_{n>=1} a(n)/A017706(n) = zeta(21).

Dirichlet g.f. of a(n)/A017706(n): zeta(s)*zeta(s+21).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017706(k) = zeta(22). (End)