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A013966 a(n) = sigma_18(n), the sum of the 18th powers of the divisors of n.

%I A013966 #45 Oct 29 2023 02:33:32
%S A013966 1,262145,387420490,68719738881,3814697265626,101560344351050,
%T A013966 1628413597910450,18014467229220865,150094635684419611,
%U A013966 1000003814697527770,5559917313492231482,26623434909949071690,112455406951957393130,426880482624234915250,1477891883850485076740
%N A013966 a(n) = sigma_18(n), the sum of the 18th powers of the divisors of n.
%C A013966 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 A013966 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 A013966 Seiichi Manyama, <a href="/A013966/b013966.txt">Table of n, a(n) for n = 1..10000</a>
%H A013966 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>.
%F A013966 G.f.: Sum_{k>=1} k^18*x^k/(1-x^k). - _Benoit Cloitre_, Apr 21 2003
%F A013966 From _Amiram Eldar_, Oct 29 2023: (Start)
%F A013966 Multiplicative with a(p^e) = (p^(18*e+18)-1)/(p^18-1).
%F A013966 Dirichlet g.f.: zeta(s)*zeta(s-18).
%F A013966 Sum_{k=1..n} a(k) = zeta(19) * n^19 / 19. + O(n^20). (End)
%t A013966 Table[DivisorSigma[18,n],{n,50}] (* _Vladimir Joseph Stephan Orlovsky_, Mar 11 2009 *)
%o A013966 (Sage) [sigma(n,18)for n in range(1,13)] # _Zerinvary Lajos_, Jun 04 2009
%o A013966 (PARI) a(n)=sigma(n,18) \\ _Charles R Greathouse IV_, Apr 28 2011
%o A013966 (Magma) [DivisorSigma(18,n): n in [1..50]]; // _G. C. Greubel_, Nov 03 2018
%Y A013966 Cf. A000203, A001157-A001160, A013677, A013954-A013972, A017665-A017712.
%K A013966 nonn,mult,easy
%O A013966 1,2
%A A013966 _N. J. A. Sloane_