A017699 Numerator of sum of -18th powers of divisors of n.

1, 262145, 387420490, 68719738881, 3814697265626, 50780172175525, 1628413597910450, 18014467229220865, 150094635684419611, 100000381469752777, 5559917313492231482, 4437239151658178615, 112455406951957393130, 213440241312117457625, 295578376770097015348

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)/A017700(n) = zeta(18) (A013676).

Dirichlet g.f. of a(n)/A017700(n): zeta(s)*zeta(s+18).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017700(k) = zeta(19) (A013677). (End)

Mathematica

Table[Numerator[DivisorSigma[18, n]/n^18], {n, 1, 20}] (* G. C. Greubel, Nov 05 2018 *)

Prog

(PARI) vector(20, n, numerator(sigma(n, 18)/n^18)) \\ G. C. Greubel, Nov 05 2018
(Magma) [Numerator(DivisorSigma(18, n)/n^18): n in [1..20]]; // G. C. Greubel, Nov 05 2018

Crossrefs

Cf. A017700 (denominator), A013676, A013677.

Sequence in context: A051441 A351315 A352984 * A013966 A036096 A170792

Adjacent sequences: A017696 A017697 A017698 * A017700 A017701 A017702

Keyword

nonn,frac

Author

N. J. A. Sloane

Status

approved