1,2

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

G. C. Greubel, Table of n, a(n) for n = 1..10000

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)

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

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

(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

Cf. A017680 (denominator), A013666, A013667.

Sequence in context: A321564 A034682 A351303 * A013956 A294303 A036086

Adjacent sequences: A017676 A017677 A017678 * A017680 A017681 A017682

nonn,frac

approved