A017667 Numerator of sum of -2nd powers of divisors of n.

1, 5, 10, 21, 26, 25, 50, 85, 91, 13, 122, 35, 170, 125, 52, 341, 290, 455, 362, 273, 500, 305, 530, 425, 651, 425, 820, 75, 842, 13, 962, 1365, 1220, 725, 52, 637, 1370, 905, 1700, 221, 1682, 625, 1850, 1281, 2366, 1325, 2210, 1705, 2451, 651, 2900, 1785

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

Numerators of coefficients in expansion of Sum_{k>=1} x^k/(k^2*(1 - x^k)). - Ilya Gutkovskiy, May 24 2018

C. Defant proves that there are no positive integers n such that sigma_{-2}(n) lies in (Pi^2/8, 5/4). See arxiv link. - Michel Marcus, Aug 24 2018

Examples

			1, 5/4, 10/9, 21/16, 26/25, 25/18, 50/49, 85/64, 91/81, 13/10, 122/121, 35/24, 170/169, ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Colin Defant, On the Density of Ranges of Generalized Divisor Functions, Notes on Number Theory and Discrete Mathematics, Vol. 21, No. 3 (2015), pp. 80-87; arXiv preprint, arXiv:1506.05432 [math.NT], 2015.

Crossrefs

Cf. A017668 (denominator), A002117, A013661, A111003 (Pi^2/8).

Programs

Magma

[Numerator(DivisorSigma(2,n)/n^2): n in [1..50]]; // G. C. Greubel, Nov 08 2018

Mathematica

Table[Numerator[DivisorSigma[-2, n]], {n, 50}] (* Vladimir Joseph Stephan Orlovsky, Jul 21 2011 *)
Table[Numerator[DivisorSigma[2, n]/n^2], {n, 1, 50}] (* G. C. Greubel, Nov 08 2018 *)

PARI

a(n) = numerator(sigma(n, -2)); \\ Michel Marcus, Aug 24 2018

PARI

vector(50, n, numerator(sigma(n, 2)/n^2)) \\ G. C. Greubel, Nov 08 2018

Formula

Dirichlet g.f.: zeta(s)*zeta(s+2) [for fraction A017667/A017668]. - Franklin T. Adams-Watters, Sep 11 2005

sup_{n>=1} a(n)/A017668(n) = zeta(2) (A013661). - Amiram Eldar, Sep 25 2022

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017668(k) = zeta(3) (A002117). - Amiram Eldar, Apr 02 2024