A017671 Numerator of sum of -4th powers of divisors of n.

1, 17, 82, 273, 626, 697, 2402, 4369, 6643, 5321, 14642, 3731, 28562, 20417, 51332, 69905, 83522, 112931, 130322, 85449, 196964, 124457, 279842, 179129, 391251, 242777, 538084, 46839, 707282, 218161, 923522, 1118481, 1200644, 41761, 1503652, 604513, 1874162

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^4*(1 - x^k)). - Ilya Gutkovskiy, May 24 2018

From Amiram Eldar, Apr 02 2024: (Start)

sup_{n>=1} a(n)/A017672(n) = zeta(4) (A013662).

Dirichlet g.f. of a(n)/A017672(n): zeta(s)*zeta(s+4).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017672(k) = zeta(5) (A013663). (End)

Example

1, 17/16, 82/81, 273/256, 626/625, 697/648, 2402/2401, 4369/4096, 6643/6561, 5321/5000, ...

Mathematica

Table[Numerator[DivisorSigma[-4, n]], {n, 50}] (* Vladimir Joseph Stephan Orlovsky, Jul 21 2011 *)
Table[Numerator[DivisorSigma[4, n]/n^4], {n, 1, 40}] (* G. C. Greubel, Nov 08 2018 *)

Prog

(PARI) vector(40, n, numerator(sigma(n, 4)/n^4)) \\ G. C. Greubel, Nov 08 2018
(Magma) [Numerator(DivisorSigma(4, n)/n^4): n in [1..40]]; // G. C. Greubel, Nov 08 2018

Crossrefs

Cf. A017672 (denominator), A013662, A013663.

Sequence in context: A321560 A034678 A065960 * A001159 A053820 A294288

Adjacent sequences: A017668 A017669 A017670 * A017672 A017673 A017674

Keyword

nonn,frac

Author

N. J. A. Sloane

Status

approved