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
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
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.
FORMULA
Dirichlet g.f.: zeta(s)*zeta(s+2) [for fraction A017667/A017668]. - Franklin T. Adams-Watters, Sep 11 2005
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017668(k) = zeta(3) (A002117). - Amiram Eldar, Apr 02 2024
EXAMPLE
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, ...
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 *)
PROG
(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
(Magma) [Numerator(DivisorSigma(2, n)/n^2): n in [1..50]]; // G. C. Greubel, Nov 08 2018
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
STATUS
approved