A017703 Numerator of sum of -20th powers of divisors of n.

1, 1048577, 3486784402, 1099512676353, 95367431640626, 1828080963947977, 79792266297612002, 1152922604119523329, 12157665462543713203, 50000047683716344601, 672749994932560009202, 638960608284819107651, 19004963774880799438802, 41834167608775550110577

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. A017704 (denominator), A013678.

Programs

Magma

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

Mathematica

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

PARI

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

Formula

From Amiram Eldar, Apr 02 2024: (Start)

sup_{n>=1} a(n)/A017704(n) = zeta(20) (A013678).

Dirichlet g.f. of a(n)/A017704(n): zeta(s)*zeta(s+20).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017704(k) = zeta(21). (End)