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

1,2

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^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)

Example

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

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 *)

Prog

(PARI) vector(20, n, numerator(sigma(n, 10)/n^10)) \\ G. C. Greubel, Nov 07 2018
(Magma) [Numerator(DivisorSigma(10, n)/n^10): n in [1..20]]; // G. C. Greubel, Nov 07 2018

Crossrefs

Cf. A017684 (denominator), A013668, A013669.

Sequence in context: A351273 A321807 A351305 * A013958 A294305 A036088

Adjacent sequences: A017680 A017681 A017682 * A017684 A017685 A017686

Keyword

nonn,frac

Author

N. J. A. Sloane

Status

approved