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%I #72 Oct 29 2023 02:36:50
%S 1,2049,177148,4196353,48828126,362976252,1977326744,8594130945,
%T 31381236757,100048830174,285311670612,743375541244,1792160394038,
%U 4051542498456,8649804864648,17600780175361,34271896307634,64300154115093,116490258898220,204900053024478
%N a(n) = sigma_11(n), the sum of the 11th 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
%C Related to congruence properties of the Ramanujan tau function since A000594(n) == a(n) (mod 691) = A046694(n). - _Benoit Cloitre_, Aug 28 2002
%H T. D. Noe, <a href="/A013959/b013959.txt">Table of n, a(n) for n=1..1000</a>
%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>.
%F G.f.: Sum_{k>=1} k^11*x^k/(1-x^k). - _Benoit Cloitre_, Apr 21 2003
%F Dirichlet g.f.: zeta(s-11)*zeta(s). - _Ilya Gutkovskiy_, Sep 10 2016
%F From _Amiram Eldar_, Oct 29 2023: (Start)
%F Multiplicative with a(p^e) = (p^(11*e+11)-1)/(p^11-1).
%F Sum_{k=1..n} a(k) = zeta(12) * n^12 / 12 + O(n^13). (End)
%t Table[DivisorSigma[11, n], {n, 30}] (* _Vincenzo Librandi_, Sep 10 2016 *)
%o (Sage) [sigma(n,11)for n in range(1,18)] # _Zerinvary Lajos_, Jun 04 2009
%o (PARI) a(n)=sigma(n,11) \\ _Charles R Greathouse IV_, Apr 28, 2011
%o (PARI) my(N=99, q='q+O('q^N)); Vec(sum(n=1, N, n^11*q^n/(1-q^n))) \\ _Altug Alkan_, Sep 10 2016
%o (Magma) [DivisorSigma(11, n): n in [1..20]]; // _Vincenzo Librandi_, Sep 10 2016
%o (Python)
%o from sympy import divisor_sigma
%o def A013959(n): return divisor_sigma(n,11) # _Chai Wah Wu_, Nov 17 2022
%Y Cf. A000594, A027860, A046694.
%Y Cf. A000203, A001157-A001160, A013670, A013954-A013972, A017665-A017712.
%K nonn,mult,easy
%O 1,2
%A _N. J. A. Sloane_
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