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A013955 a(n) = sigma_7(n), the sum of the 7th powers of the divisors of n. 24

%I

%S 1,129,2188,16513,78126,282252,823544,2113665,4785157,10078254,

%T 19487172,36130444,62748518,106237176,170939688,270549121,410338674,

%U 617285253,893871740,1290094638,1801914272,2513845188

%N a(n) = sigma_7(n), the sum of the 7th 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

%D Max Koecher and Aloys Krieg, Elliptische Funktionen und Modulformen, 2. Auflage, Springer, 2007, p. 51.

%D J.-P. Serre, A Course in Arithmetic, Springer-Verlag, 1973, Chap. VII, Section 4., p. 93.

%H Vincenzo Librandi, <a href="/A013955/b013955.txt">Table of n, a(n) for n = 1..1000</a>

%H 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 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>

%F 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 G.f.: sum_{k>=1} k^7*x^k/(1-x^k). - _Benoit Cloitre_, Apr 21 2003

%F 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

%t lst={};Do[AppendTo[lst,DivisorSigma[7,n]],{n,5!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Mar 11 2009 *)

%t DivisorSigma[7,Range[30]] (* _Harvey P. Dale_, Dec 10 2016 *)

%o (PARI) a(n)=if(n<1,0,sigma(n,7))

%o (Sage) [sigma(n,7)for n in xrange(1,23)] # _Zerinvary Lajos_, Jun 04 2009

%o (MAGMA) [DivisorSigma(7,n): n in [1..30]]; // _Bruno Berselli_, Apr 10 2013

%K nonn,mult

%O 1,2

%A _N. J. A. Sloane_

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Last modified October 20 02:18 EDT 2019. Contains 328244 sequences. (Running on oeis4.)