%I #38 Oct 29 2023 08:36:37

%S 1,1048577,3486784402,1099512676353,95367431640626,3656161927895954,

%T 79792266297612002,1152922604119523329,12157665462543713203,

%U 100000095367432689202,672749994932560009202,3833763649708914645906,19004963774880799438802,83668335217551100221154

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

%C 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 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 Seiichi Manyama, <a href="/A013968/b013968.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>.

%F G.f.: Sum_{k>=1} k^20*x^k/(1-x^k). - _Benoit Cloitre_, Apr 21 2003

%F From _Amiram Eldar_, Oct 29 2023: (Start)

%F Multiplicative with a(p^e) = (p^(20*e+20)-1)/(p^20-1).

%F Dirichlet g.f.: zeta(s)*zeta(s-20).

%F Sum_{k=1..n} a(k) = zeta(21) * n^21 / 21 + O(n^22). (End)

%t DivisorSigma[20,Range[20]] (* _Harvey P. Dale_, Jul 26 2015 *)

%o (Sage) [sigma(n,20)for n in range(1,13)] # _Zerinvary Lajos_, Jun 04 2009

%o (PARI) vector(50, n, sigma(n,20)) \\ _G. C. Greubel_, Nov 03 2018

%o (Magma) [DivisorSigma(20,n): n in [1..50]]; // _G. C. Greubel_, Nov 03 2018

%Y Cf. A000203, A001157-A001160, A013954-A013972, A017665-A017712.

%K nonn,easy,mult

%O 1,2

%A _N. J. A. Sloane_