A017677 Numerator of sum of -7th powers of divisors of n.

1, 129, 2188, 16513, 78126, 23521, 823544, 2113665, 4785157, 5039127, 19487172, 9032611, 62748518, 13279647, 56979896, 270549121, 410338674, 205761751, 893871740, 645047319, 1801914272, 628461297, 3404825448, 385391585, 6103593751, 4047279411, 10465138360, 34691791

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^7*(1 - x^k)). - Ilya Gutkovskiy, May 25 2018

From Amiram Eldar, Apr 02 2024: (Start)

sup_{n>=1} a(n)/A017678(n) = zeta(7) (A013665).

Dirichlet g.f. of a(n)/A017678(n): zeta(s)*zeta(s+7).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017678(k) = zeta(8) (A013666). (End)

Example

1, 129/128, 2188/2187, 16513/16384, 78126/78125, 23521/23328, 823544/823543, 2113665/2097152, ...

Mathematica

Table[Numerator[Total[Divisors[n]^-7]], {n, 30}] (* Harvey P. Dale, Nov 29 2014 *)
Table[Numerator[DivisorSigma[7, n]/n^7], {n, 1, 20}] (* G. C. Greubel, Nov 07 2018 *)

Prog

(PARI) vector(20, n, numerator(sigma(n, 7)/n^7)) \\ G. C. Greubel, Nov 07 2018
(Magma) [Numerator(DivisorSigma(7, n)/n^7): n in [1..20]]; // G. C. Greubel, Nov 07 2018

Crossrefs

Cf. A017678 (denominator), A013665, A013666.

Sequence in context: A321563 A034681 A351302 * A013955 A294302 A343509

Adjacent sequences: A017674 A017675 A017676 * A017678 A017679 A017680

Keyword

nonn,frac

Author

N. J. A. Sloane

Status

approved