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%I A013955 #65 Oct 29 2023 02:34:55
%S A013955 1,129,2188,16513,78126,282252,823544,2113665,4785157,10078254,
%T A013955 19487172,36130444,62748518,106237176,170939688,270549121,410338674,
%U A013955 617285253,893871740,1290094638,1801914272,2513845188,3404825448,4624699020,6103593751,8094558822,10465138360
%N A013955 a(n) = sigma_7(n), the sum of the 7th powers of the divisors of n.
%C A013955 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 A013955 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
%D A013955 Max Koecher and Aloys Krieg, Elliptische Funktionen und Modulformen, 2. Auflage, Springer, 2007, p. 51.
%D A013955 Jean-Pierre Serre, A Course in Arithmetic, Springer-Verlag, 1973, Chap. VII, Section 4., p. 93.
%H A013955 Vincenzo Librandi, <a href="/A013955/b013955.txt">Table of n, a(n) for n = 1..1000</a>
%H A013955 D. B. Lahiri, <a href="https://doi.org/10.1017/S0004972700042179">Some arithmetical identities for Ramanujan's and divisor functions</a>, Bulletin of the Australian Mathematical Society, Volume 1, Issue 3 December 1969, pp. 307-314. See Theorem 3 p. 308.
%H A013955 Don Zagier, <a href="http://people.mpim-bonn.mpg.de/zagier/files/doi/10.1007/978-3-540-74119-0_1/fulltext.pdf">Elliptic modular forms and their applications</a>, The 1-2-3 of modular forms. Springer Berlin Heidelberg, 2008. 1-103. See p. 17, G_8(z).
%H A013955 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>.
%F A013955 Let sigma(p,n) be the sum of the p-th powers of the divisors of n. Then sigma(7,n) = sigma(3,n) + 120 sum(sigma(3,k) sigma(3,n-k),k=1..n-1) (Cf. A087115). - Eugene Salamin, Apr 29 2006 [Hurwitz Identity, Math. Werke I, 1-66, p. 50, last line. See, e.g., the Koecher-Krieg reference, p. 51, rewritten. - _Wolfdieter Lang_, Jan 20 2016]
%F A013955 G.f.: Sum_{k>=1} k^7*x^k/(1-x^k). - _Benoit Cloitre_, Apr 21 2003
%F A013955 L.g.f.: -log(Product_{k>=1} (1 - x^k)^(k^6)) = Sum_{n>=1} a(n)*x^n/n. - _Ilya Gutkovskiy_, May 06 2017
%F A013955 From _Amiram Eldar_, Oct 29 2023: (Start)
%F A013955 Multiplicative with a(p^e) = (p^(7*e+7)-1)/(p^7-1).
%F A013955 Dirichlet g.f.: zeta(s)*zeta(s-7).
%F A013955 Sum_{k=1..n} a(k) = zeta(8) * n^8 / 8 + O(n^9). (End)
%t A013955 lst={};Do[AppendTo[lst,DivisorSigma[7,n]],{n,5!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Mar 11 2009 *)
%t A013955 DivisorSigma[7,Range[30]] (* _Harvey P. Dale_, Dec 10 2016 *)
%o A013955 (PARI) a(n)=if(n<1,0,sigma(n,7))
%o A013955 (Sage) [sigma(n, 7) for n in range(1, 23)]  # _Zerinvary Lajos_, Jun 04 2009
%o A013955 (Magma) [DivisorSigma(7,n): n in [1..30]]; // _Bruno Berselli_, Apr 10 2013
%Y A013955 Cf. A000203, A001157-A001160, A013666, A013954-A013972, A017665-A017712, A087115.
%K A013955 nonn,easy,mult
%O A013955 1,2
%A A013955 _N. J. A. Sloane_

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