A017683 Numerator of sum of -10th powers of divisors of n.

1, 1025, 59050, 1049601, 9765626, 30263125, 282475250, 1074791425, 3486843451, 200195333, 25937424602, 10329823175, 137858491850, 144768565625, 23066408612, 1100586419201, 2015993900450, 3574014537275, 6131066257802, 5125005407613, 16680163512500, 13292930108525

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

Examples

			1, 1025/1024, 59050/59049, 1049601/1048576, 9765626/9765625, 30263125/30233088, 282475250/282475249, ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A017684 (denominator), A013668, A013669.

Programs

Magma

[Numerator(DivisorSigma(10,n)/n^10): n in [1..20]]; // G. C. Greubel, Nov 07 2018

Mathematica

Table[Numerator[Total[Divisors[n]^-10]],{n,20}] (* Harvey P. Dale, Sep 04 2018 *)
Table[Numerator[DivisorSigma[10, n]/n^10], {n, 1, 20}] (* G. C. Greubel, Nov 07 2018 *)

PARI

vector(20, n, numerator(sigma(n, 10)/n^10)) \\ G. C. Greubel, Nov 07 2018

Formula

Numerators of coefficients in expansion of Sum_{k>=1} x^k/(k^10*(1 - x^k)). - Ilya Gutkovskiy, May 25 2018

From Amiram Eldar, Apr 02 2024: (Start)

sup_{n>=1} a(n)/A017684(n) = zeta(10) (A013668).

Dirichlet g.f. of a(n)/A017684(n): zeta(s)*zeta(s+10).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017684(k) = zeta(11) (A013669). (End)