A017707 Numerator of sum of -22nd powers of divisors of n.

1, 4194305, 31381059610, 17592190238721, 2384185791015626, 65810867613760525, 3909821048582988050, 73786993887028445185, 984770902214992292491, 1000000238418579520993, 81402749386839761113322, 92010261758627305193135, 3211838877954855105157370, 8199490986588434846527625

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

From Amiram Eldar, Apr 02 2024: (Start)

sup_{n>=1} a(n)/A017708(n) = zeta(22).

Dirichlet g.f. of a(n)/A017708(n): zeta(s)*zeta(s+22).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017708(k) = zeta(23). (End)

Mathematica

Table[Numerator[DivisorSigma[22, n]/n^22], {n, 1, 20}] (* G. C. Greubel, Nov 05 2018 *)
DivisorSigma[-22, Range[20]]//Numerator (* Harvey P. Dale, Sep 19 2023 *)

Prog

(PARI) vector(20, n, numerator(sigma(n, 22)/n^22)) \\ G. C. Greubel, Nov 05 2018
(Magma) [Numerator(DivisorSigma(22, n)/n^22): n in [1..20]]; // G. C. Greubel, Nov 05 2018

Crossrefs

Cf. A017708 (denominator).

Sequence in context: A017327 A017447 A017579 * A013970 A036100 A236946

Adjacent sequences: A017704 A017705 A017706 * A017708 A017709 A017710

Keyword

nonn,frac

Author

N. J. A. Sloane

Status

approved