A017673 Numerator of sum of -5th powers of divisors of n.

1, 33, 244, 1057, 3126, 671, 16808, 33825, 59293, 51579, 161052, 64477, 371294, 69333, 254248, 1082401, 1419858, 652223, 2476100, 1652091, 4101152, 120789, 6436344, 687775, 9768751, 6126351, 14408200, 317251, 20511150, 349591, 28629152, 34636833, 13098896

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

Numerators of coefficients in expansion of Sum_{k>=1} x^k/(k^5*(1 - x^k)). - Ilya Gutkovskiy, May 25 2018

From Amiram Eldar, Apr 02 2024: (Start)

sup_{n>=1} a(n)/A017674(n) = zeta(5) (A013663).

Dirichlet g.f. of a(n)/A017674(n): zeta(s)*zeta(s+5).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017674(k) = zeta(6) (A013664). (End)

Example

1, 33/32, 244/243, 1057/1024, 3126/3125, 671/648, 16808/16807, 33825/32768, 59293/59049, ...

Mathematica

Table[Numerator[DivisorSigma[-5, n]], {n, 100}] (* Vladimir Joseph Stephan Orlovsky, Jul 22 2011 *)
Table[Numerator[DivisorSigma[5, n]/n^5], {n, 1, 40}] (* G. C. Greubel, Nov 08 2018 *)

Prog

(PARI) vector(40, n, numerator(sigma(n, 5)/n^5)) \\ G. C. Greubel, Nov 08 2018
(Magma) [Numerator(DivisorSigma(5, n)/n^5): n in [1..40]]; // G. C. Greubel, Nov 08 2018

Crossrefs

Cf. A017674 (denominator), A013663, A013664.

Sequence in context: A321561 A034679 A351300 * A001160 A294300 A271208

Adjacent sequences: A017670 A017671 A017672 * A017674 A017675 A017676

Keyword

nonn,frac

Author

N. J. A. Sloane

Status

approved