This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A013961 #57 Jan 15 2025 04:18:38
%S A013961 1,8193,1594324,67117057,1220703126,13062296532,96889010408,
%T A013961 549822930945,2541867422653,10001220711318,34522712143932,
%U A013961 107006334784468,302875106592254,793811662272744,1946196290656824,4504149450301441,9904578032905938,20825519793796029,42052983462257060
%N A013961 a(n) = sigma_13(n), the sum of the 13th powers of the divisors of n.
%C A013961 If the canonical factorization of n into prime powers is the product of p^e(p) then sigma_k(n) = Product_{p prime} ((p^((e(p)+1)*k)) - 1)/(p^k - 1).
%C A013961 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
%C A013961 By Fermat's little theorem n^13 == n (mod 13). Hence sigma_13(n) == sigma_(1) (mod 13). In fact, sigma_13(n) == sigma_(1) (mod 2730), where 2730 = 2*3*5*7*13 = the numerator of Bernoulli(12). - _Peter Bala_, Jan 12 2025
%H A013961 Vincenzo Librandi, <a href="/A013961/b013961.txt">Table of n, a(n) for n = 1..1000</a>
%H A013961 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>.
%F A013961 G.f.: Sum_{k>=1} k^13*x^k/(1-x^k). - _Benoit Cloitre_, Apr 21 2003
%F A013961 Dirichlet g.f.: zeta(s-13)*zeta(s). - _Ilya Gutkovskiy_, Sep 10 2016
%F A013961 Empirical: Sum_{n>=1} a(n)/exp(2*Pi*n) = 1/24. - _Simon Plouffe_, Mar 01 2021
%F A013961 From _Amiram Eldar_, Oct 29 2023: (Start)
%F A013961 Multiplicative with a(p^e) = (p^(13*e+13)-1)/(p^13-1).
%F A013961 Sum_{k=1..n} a(k) = zeta(14) * n^14 / 14 + O(n^15). (End)
%p A013961 A013961 := proc(n)
%p A013961 numtheory[sigma][13](n) ;
%p A013961 end proc: # _R. J. Mathar_, Sep 21 2017
%t A013961 DivisorSigma[13, Range[30]] (* _Vincenzo Librandi_, Sep 10 2016 *)
%o A013961 (Sage) [sigma(n,13)for n in range(1,16)] # _Zerinvary Lajos_, Jun 04 2009
%o A013961 (Magma) [DivisorSigma(13, n): n in [1..20]]; // _Vincenzo Librandi_, Sep 10 2016
%o A013961 (PARI) my(N=99, q='q+O('q^N)); Vec(sum(n=1, N, n^13*q^n/(1-q^n))) \\ _Altug Alkan_, Sep 10 2016
%o A013961 (PARI) a(n) = sigma(n, 13); \\ _Michel Marcus_, Sep 10 2016
%Y A013961 Cf. A000203, A001157-A001160, A013672, A013954-A013972, A017665-A017712.
%K A013961 nonn,mult,easy
%O A013961 1,2
%A A013961 _N. J. A. Sloane_