A017681 Numerator of sum of -9th powers of divisors of n.

1, 513, 19684, 262657, 1953126, 93499, 40353608, 134480385, 387440173, 500976819, 2357947692, 1292535097, 10604499374, 2587675113, 1423901192, 68853957121, 118587876498, 7361363287, 322687697780, 256501107891, 113474345696, 302406791499, 1801152661464, 24510295355

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

Numerators of coefficients in expansion of Sum_{k>=1} x^k/(k^9*(1 - x^k)). - Ilya Gutkovskiy, May 25 2018

From Amiram Eldar, Apr 02 2024: (Start)

sup_{n>=1} a(n)/A017682(n) = zeta(9) (A013667).

Dirichlet g.f. of a(n)/A017682(n): zeta(s)*zeta(s+9).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017682(k) = zeta(10) (A013668). (End)

Example

1, 513/512, 19684/19683, 262657/262144, 1953126/1953125, 93499/93312, 40353608/40353607, 134480385/134217728, ...

Mathematica

Table[Numerator[Total[1/Divisors[n]^9]], {n, 20}] (* Harvey P. Dale, Aug 26 2013 *)
Table[Numerator[DivisorSigma[9, n]/n^9], {n, 1, 20}] (* G. C. Greubel, Nov 07 2018 *)

Prog

(PARI) vector(20, n, numerator(sigma(n, 9)/n^9)) \\ G. C. Greubel, Nov 07 2018
(Magma) [Numerator(DivisorSigma(9, n)/n^9): n in [1..20]]; // G. C. Greubel, Nov 07 2018

Crossrefs

Cf. A017682 (denominator), A013667, A013668.

Sequence in context: A351272 A321565 A351304 * A013957 A294304 A036087

Adjacent sequences: A017678 A017679 A017680 * A017682 A017683 A017684

Keyword

nonn,frac

Author

N. J. A. Sloane

Status

approved