A017701 Numerator of sum of -19th powers of divisors of n.

1, 524289, 1162261468, 274878431233, 19073486328126, 50780075233021, 11398895185373144, 144115462954287105, 1350851718835253557, 5000009536743426207, 61159090448414546292, 79870152251600907511, 1461920290375446110678, 747039419730512536827, 7389459406535218149656

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)/A017702(n) = zeta(19) (A013677).

Dirichlet g.f. of a(n)/A017702(n): zeta(s)*zeta(s+19).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017702(k) = zeta(20) (A013678). (End)

Mathematica

Table[Numerator[DivisorSigma[19, n]/n^19], {n, 1, 20}] (* G. C. Greubel, Nov 05 2018 *)

Prog

(PARI) vector(20, n, numerator(sigma(n, 19)/n^19)) \\ G. C. Greubel, Nov 05 2018
(Magma) [Numerator(DivisorSigma(19, n)/n^19): n in [1..20]]; // G. C. Greubel, Nov 05 2018

Crossrefs

Cf. A017702 (denominator), A013677, A013678.

Sequence in context: A010807 A236227 A320345 * A013967 A036097 A254039

Adjacent sequences: A017698 A017699 A017700 * A017702 A017703 A017704

Keyword

nonn,frac

Author

N. J. A. Sloane

Status

approved