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%I #39 Oct 29 2023 02:38:19
%S 1,32769,14348908,1073774593,30517578126,470199366252,4747561509944,
%T 35185445863425,205891146443557,1000030517610894,4177248169415652,
%U 15407492847694444,51185893014090758,155572843119354936,437893920912786408,1152956690052710401,2862423051509815794
%N a(n) = sigma_15(n), the sum of the 15th 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 Vincenzo Librandi, <a href="/A013963/b013963.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^15*x^k/(1-x^k). - _Benoit Cloitre_, Apr 21 2003
%F Dirichlet g.f.: zeta(s-15)*zeta(s). - _Ilya Gutkovskiy_, Sep 10 2016
%F From _Amiram Eldar_, Oct 29 2023: (Start)
%F Multiplicative with a(p^e) = (p^(15*e+15)-1)/(p^15-1).
%F Sum_{k=1..n} a(k) = zeta(16) * n^16 / 16 + O(n^17). (End)
%t DivisorSigma[15, Range[30]] (* _Vincenzo Librandi_, Sep 10 2016 *)
%o (Sage) [sigma(n,15)for n in range(1,15)] # _Zerinvary Lajos_, Jun 04 2009
%o (Magma) [DivisorSigma(15, n): n in [1..20]]; // _Vincenzo Librandi_, Sep 10 2016
%o (PARI) my(N=99, q='q+O('q^N)); Vec(sum(n=1, N, n^15*q^n/(1-q^n))) \\ _Altug Alkan_, Sep 10 2016
%o (PARI) a(n) = sigma(n, 15); \\ _Amiram Eldar_, Oct 29 2023
%Y Cf. A000203, A001157-A001160, A013674, A013954-A013972, A017665-A017712.
%K nonn,mult,easy
%O 1,2
%A _N. J. A. Sloane_
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