A017679 Numerator of sum of -8th powers of divisors of n.

1, 257, 6562, 65793, 390626, 843217, 5764802, 16843009, 43053283, 50195441, 214358882, 71955611, 815730722, 740777057, 2563287812, 4311810305, 6975757442, 11064693731, 16983563042, 12850228209, 37828630724, 27545116337, 78310985282, 55261912529, 152588281251

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^8*(1 - x^k)). - Ilya Gutkovskiy, May 25 2018

From Amiram Eldar, Apr 02 2024: (Start)

sup_{n>=1} a(n)/A017680(n) = zeta(8) (A013666).

Dirichlet g.f. of a(n)/A017680(n): zeta(s)*zeta(s+8).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017680(k) = zeta(9) (A013667). (End)

Example

1, 257/256, 6562/6561, 65793/65536, 390626/390625, 843217/839808, 5764802/5764801, 16843009/16777216, ...

Mathematica

Table[Numerator[DivisorSigma[8, n]/n^8], {n, 1, 20}] (* G. C. Greubel, Nov 07 2018 *)

Prog

(PARI) vector(20, n, numerator(sigma(n, 8)/n^8)) \\ G. C. Greubel, Nov 07 2018
(Magma) [Numerator(DivisorSigma(8, n)/n^8): n in [1..20]]; // G. C. Greubel, Nov 07 2018

Crossrefs

Cf. A017680 (denominator), A013666, A013667.

Sequence in context: A321564 A034682 A351303 * A013956 A294303 A036086

Adjacent sequences: A017676 A017677 A017678 * A017680 A017681 A017682

Keyword

nonn,frac

Author

N. J. A. Sloane

Status

approved