A017685 Numerator of sum of -11th powers of divisors of n.

1, 2049, 177148, 4196353, 48828126, 30248021, 1977326744, 8594130945, 31381236757, 50024415087, 285311670612, 185843885311, 1792160394038, 506442812307, 2883268288216, 17600780175361, 34271896307634, 21433384705031, 116490258898220, 102450026512239, 350279478046112

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)/A017686(n) = zeta(11) (A013669).

Dirichlet g.f. of a(n)/A017686(n): zeta(s)*zeta(s+11).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017686(k) = zeta(12) (A013670). (End)

Mathematica

Table[Numerator[Total[Divisors[n]^-11]], {n, 20}] (* Harvey P. Dale, Aug 26 2012 *)
Table[Numerator[DivisorSigma[11, n]/n^11], {n, 1, 20}] (* G. C. Greubel, Nov 06 2018 *)

Prog

(PARI) vector(20, n, numerator(sigma(n, 11)/n^11)) \\ G. C. Greubel, Nov 06 2018
(Magma) [Numerator(DivisorSigma(11, n)/n^11): n in [1..20]]; // G. C. Greubel, Nov 06 2018

Crossrefs

Cf. A017686 (denominator), A013669, A013670.

Sequence in context: A060949 A230189 A321808 * A013959 A036089 A123095

Adjacent sequences: A017682 A017683 A017684 * A017686 A017687 A017688

Keyword

nonn,frac

Author

N. J. A. Sloane

Status

approved