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

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. A017686 (denominator), A013669, A013670.

Programs

Magma

[Numerator(DivisorSigma(11,n)/n^11): n in [1..20]]; // G. C. Greubel, Nov 06 2018

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 *)

PARI

vector(20, n, numerator(sigma(n, 11)/n^11)) \\ G. C. Greubel, Nov 06 2018

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)