A017669 Numerator of sum of -3rd powers of divisors of n.

1, 9, 28, 73, 126, 7, 344, 585, 757, 567, 1332, 511, 2198, 387, 392, 4681, 4914, 757, 6860, 4599, 1376, 2997, 12168, 455, 15751, 9891, 20440, 3139, 24390, 147, 29792, 37449, 4144, 22113, 6192, 55261, 50654, 15435, 61544, 7371, 68922, 172, 79508, 24309, 10598

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, 9/8, 28/27, 73/64, 126/125, 7/6, 344/343, 585/512, 757/729, 567/500, 1332/1331, 511/432, ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Vincenzo Librandi)

Crossrefs

Cf. A017670 (denominator), A002117, A013662.

Programs

Magma

[Numerator(DivisorSigma(3,n)/n^3): n in [1..40]]; // G. C. Greubel, Nov 08 2018

Mathematica

Table[Numerator[DivisorSigma[-3, n]], {n, 50}] (* Vladimir Joseph Stephan Orlovsky, Jul 21 2011 *)
Table[Numerator[DivisorSigma[3, n]/n^3], {n, 1, 40}] (* G. C. Greubel, Nov 08 2018 *)

PARI

vector(40, n, numerator(sigma(n, 3)/n^3)) \\ G. C. Greubel, Nov 08 2018

Formula

Numerators of coefficients in expansion of Sum_{k>=1} x^k/(k^3*(1 - x^k)). - Ilya Gutkovskiy, May 24 2018

From Amiram Eldar, Apr 02 2024: (Start)

sup_{n>=1} a(n)/A017670(n) = zeta(3) (A002117).

Dirichlet g.f. of a(n)/A017670(n): zeta(s)*zeta(s+3).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017670(k) = zeta(4) (A013662). (End)