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A013967 a(n) = sigma_19(n), the sum of the 19th powers of the divisors of n.

%I A013967 #34 Oct 29 2023 02:39:13
%S A013967 1,524289,1162261468,274878431233,19073486328126,609360902796252,
%T A013967 11398895185373144,144115462954287105,1350851718835253557,
%U A013967 10000019073486852414,61159090448414546292,319480609006403630044,1461920290375446110678,5976315357844100294616
%N A013967 a(n) = sigma_19(n), the sum of the 19th powers of the divisors of n.
%C A013967 If the canonical factorization of n into prime powers is the product of p^e(p) then sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1).
%C A013967 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 A013967 Seiichi Manyama, <a href="/A013967/b013967.txt">Table of n, a(n) for n = 1..10000</a>
%H A013967 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>.
%F A013967 G.f.: Sum_{k>=1} k^19*x^k/(1-x^k). - _Benoit Cloitre_, Apr 21 2003
%F A013967 From _Amiram Eldar_, Oct 29 2023: (Start)
%F A013967 Multiplicative with a(p^e) = (p^(19*e+19)-1)/(p^19-1).
%F A013967 Dirichlet g.f.: zeta(s)*zeta(s-19).
%F A013967 Sum_{k=1..n} a(k) = zeta(20) * n^20 / 20 + O(n^21). (End)
%t A013967 Table[DivisorSigma[19,n],{n,50}] (* _Vladimir Joseph Stephan Orlovsky_, Mar 11 2009 *)
%o A013967 (Sage) [sigma(n,19)for n in range(1,13)] # _Zerinvary Lajos_, Jun 04 2009
%o A013967 (PARI) vector(50, n, sigma(n,19)) \\ _G. C. Greubel_, Nov 03 2018
%o A013967 (Magma) [DivisorSigma(19,n): n in [1..50]]; // _G. C. Greubel_, Nov 03 2018
%Y A013967 Cf. A000203, A001157-A001160, A013678, A013954-A013972, A017665-A017712.
%K A013967 nonn,easy,mult
%O A013967 1,2
%A A013967 _N. J. A. Sloane_