cp's OEIS Frontend

A017673 Numerator of sum of -5th powers of divisors of n.

%I A017673 #24 Apr 02 2024 02:55:43
%S A017673 1,33,244,1057,3126,671,16808,33825,59293,51579,161052,64477,371294,
%T A017673 69333,254248,1082401,1419858,652223,2476100,1652091,4101152,120789,
%U A017673 6436344,687775,9768751,6126351,14408200,317251,20511150,349591,28629152,34636833,13098896
%N A017673 Numerator of sum of -5th powers of divisors of n.
%C A017673 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
%H A017673 G. C. Greubel, <a href="/A017673/b017673.txt">Table of n, a(n) for n = 1..10000</a>
%F A017673 Numerators of coefficients in expansion of Sum_{k>=1} x^k/(k^5*(1 - x^k)). - _Ilya Gutkovskiy_, May 25 2018
%F A017673 From _Amiram Eldar_, Apr 02 2024: (Start)
%F A017673 sup_{n>=1} a(n)/A017674(n) = zeta(5) (A013663).
%F A017673 Dirichlet g.f. of a(n)/A017674(n): zeta(s)*zeta(s+5).
%F A017673 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017674(k) = zeta(6) (A013664). (End)
%e A017673 1, 33/32, 244/243, 1057/1024, 3126/3125, 671/648, 16808/16807, 33825/32768, 59293/59049, ...
%t A017673 Table[Numerator[DivisorSigma[-5, n]], {n,100}] (* _Vladimir Joseph Stephan Orlovsky_, Jul 22 2011 *)
%t A017673 Table[Numerator[DivisorSigma[5, n]/n^5], {n, 1, 40}] (* _G. C. Greubel_, Nov 08 2018 *)
%o A017673 (PARI) vector(40, n, numerator(sigma(n, 5)/n^5)) \\ _G. C. Greubel_, Nov 08 2018
%o A017673 (Magma) [Numerator(DivisorSigma(5,n)/n^5): n in [1..40]]; // _G. C. Greubel_, Nov 08 2018
%Y A017673 Cf. A017674 (denominator), A013663, A013664.
%K A017673 nonn,frac
%O A017673 1,2
%A A017673 _N. J. A. Sloane_