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%I #44 Oct 29 2023 02:57:29
%S 1,257,6562,65793,390626,1686434,5764802,16843009,43053283,100390882,
%T 214358882,431733666,815730722,1481554114,2563287812,4311810305,
%U 6975757442,11064693731,16983563042,25700456418,37828630724,55090232674,78310985282,110523825058,152588281251
%N a(n) = sigma_8(n), the sum of the 8th 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="/A013956/b013956.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^8*x^k/(1-x^k). - _Benoit Cloitre_, Apr 21 2003
%F L.g.f.: -log(Product_{k>=1} (1 - x^k)^(k^7)) = Sum_{n>=1} a(n)*x^n/n. - _Ilya Gutkovskiy_, May 06 2017
%F From _Amiram Eldar_, Oct 29 2023: (Start)
%F Multiplicative with a(p^e) = (p^(8*e+8)-1)/(p^8-1).
%F Dirichlet g.f.: zeta(s)*zeta(s-8).
%F Sum_{k=1..n} a(k) = zeta(9) * n^9 / 9 + O(n^10). (End)
%t Table[DivisorSigma[8,n],{n,50}] (* _Vladimir Joseph Stephan Orlovsky_, Mar 11 2009 *)
%o (Sage) [sigma(n,8)for n in range(1,21)] # _Zerinvary Lajos_, Jun 04 2009
%o (PARI) a(n)=sigma(n,8) \\ _Charles R Greathouse IV_, Apr 28, 2011
%o (Magma) [DivisorSigma(8,n): n in [1..30]]; // _Bruno Berselli_, Apr 10 2013
%Y Cf. A000203, A001157-A001160, A013667, A013954-A013972, A017665-A017712.
%K nonn,mult,easy
%O 1,2
%A _N. J. A. Sloane_
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