A017687 Numerator of sum of -12th powers of divisors of n.

1, 4097, 531442, 16781313, 244140626, 1088658937, 13841287202, 68736258049, 282430067923, 500122072361, 3138428376722, 1486382423891, 23298085122482, 28353876833297, 129746582562692, 281543712968705, 582622237229762, 1157115988280531, 2213314919066162, 2048500130460969

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)/A017688(n) = zeta(12) (A013670).

Dirichlet g.f. of a(n)/A017688(n): zeta(s)*zeta(s+12).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017688(k) = zeta(13) (A013671). (End)

Mathematica

Table[Numerator[DivisorSigma[12, n]/n^12], {n, 1, 20}] (* G. C. Greubel, Nov 06 2018 *)

Prog

(PARI) vector(20, n, numerator(sigma(n, 12)/n^12)) \\ G. C. Greubel, Nov 06 2018
(Magma) [Numerator(DivisorSigma(12, n)/n^12): n in [1..20]]; // G. C. Greubel, Nov 06 2018

Crossrefs

Cf. A017688 (denominator), A013670, A013671.

Sequence in context: A342685 A342686 A321809 * A013960 A036090 A123094

Adjacent sequences: A017684 A017685 A017686 * A017688 A017689 A017690

Keyword

nonn,frac

Author

N. J. A. Sloane

Status

approved