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A013972 a(n) = sigma_24(n), the sum of the 24th powers of the divisors of n.

%I A013972 #51 Jun 29 2025 05:29:35
%S A013972 1,16777217,282429536482,281474993487873,59604644775390626,
%T A013972 4738381620767930594,191581231380566414402,4722366764344638701569,
%U A013972 79766443077154939399843,1000000059604644792167842,9849732675807611094711842,79496851942053939878082786,542800770374370512771595362
%N A013972 a(n) = sigma_24(n), the sum of the 24th powers of the divisors of n.
%C A013972 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 A013972 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 A013972 Seiichi Manyama, <a href="/A013972/b013972.txt">Table of n, a(n) for n = 1..10000</a>
%H A013972 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>.
%F A013972 G.f.: Sum_{k>=1} k^24*x^k/(1-x^k). - _Benoit Cloitre_, Apr 21 2003
%F A013972 From _Amiram Eldar_, Oct 29 2023: (Start)
%F A013972 Multiplicative with a(p^e) = (p^(24*e+24)-1)/(p^24-1).
%F A013972 Dirichlet g.f.: zeta(s)*zeta(s-24).
%F A013972 Sum_{k=1..n} a(k) = zeta(25) * n^25 / 25 + O(n^26). (End)
%t A013972 Table[DivisorSigma[24,n],{n,50}] (* _Vladimir Joseph Stephan Orlovsky_, Mar 11 2009 *)
%o A013972 (Sage) [sigma(n,24)for n in range(1,12)] # _Zerinvary Lajos_, Jun 04 2009
%o A013972 (PARI) a(n)=sigma(n,24) \\ _Charles R Greathouse IV_, Apr 28 2011
%o A013972 (Magma) [DivisorSigma(24,n): n in [1..50]]; // _G. C. Greubel_, Nov 03 2018
%Y A013972 Cf. A000203, A001157-A001160, A013954-A013972, A017665-A017712.
%K A013972 nonn,mult,easy
%O A013972 1,2
%A A013972 _N. J. A. Sloane_