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A013968 a(n) = sigma_20(n), the sum of the 20th powers of the divisors of n.

%I A013968 #38 Oct 29 2023 08:36:37
%S A013968 1,1048577,3486784402,1099512676353,95367431640626,3656161927895954,
%T A013968 79792266297612002,1152922604119523329,12157665462543713203,
%U A013968 100000095367432689202,672749994932560009202,3833763649708914645906,19004963774880799438802,83668335217551100221154
%N A013968 a(n) = sigma_20(n), the sum of the 20th powers of the divisors of n.
%C A013968 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 A013968 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 A013968 Seiichi Manyama, <a href="/A013968/b013968.txt">Table of n, a(n) for n = 1..10000</a>
%H A013968 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>.
%F A013968 G.f.: Sum_{k>=1} k^20*x^k/(1-x^k). - _Benoit Cloitre_, Apr 21 2003
%F A013968 From _Amiram Eldar_, Oct 29 2023: (Start)
%F A013968 Multiplicative with a(p^e) = (p^(20*e+20)-1)/(p^20-1).
%F A013968 Dirichlet g.f.: zeta(s)*zeta(s-20).
%F A013968 Sum_{k=1..n} a(k) = zeta(21) * n^21 / 21 + O(n^22). (End)
%t A013968 DivisorSigma[20,Range[20]] (* _Harvey P. Dale_, Jul 26 2015 *)
%o A013968 (Sage) [sigma(n,20)for n in range(1,13)] # _Zerinvary Lajos_, Jun 04 2009
%o A013968 (PARI) vector(50, n, sigma(n,20)) \\ _G. C. Greubel_, Nov 03 2018
%o A013968 (Magma) [DivisorSigma(20,n): n in [1..50]]; // _G. C. Greubel_, Nov 03 2018
%Y A013968 Cf. A000203, A001157-A001160, A013954-A013972, A017665-A017712.
%K A013968 nonn,easy,mult
%O A013968 1,2
%A A013968 _N. J. A. Sloane_