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%I M5240 N2279 #67 Jan 23 2024 02:10:50
%S 1,33,244,1057,3126,8052,16808,33825,59293,103158,161052,257908,
%T 371294,554664,762744,1082401,1419858,1956669,2476100,3304182,4101152,
%U 5314716,6436344,8253300,9768751,12252702,14408200,17766056,20511150
%N sigma_5(n), the sum of the 5th 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
%C Empirical: Sum_{n>=1} a(n)/exp(2*Pi*n) = 1/504. - _Simon Plouffe_, Mar 01 2021
%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 827.
%D G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, p. 166.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D Zagier, Don. "Elliptic modular forms and their applications." The 1-2-3 of modular forms. Springer Berlin Heidelberg, 2008. 1-103. See p. 17, G_6(z).
%H T. D. Noe, <a href="/A001160/b001160.txt">Table of n, a(n) for n = 1..10000</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 827.
%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F Multiplicative with a(p^e) = (p^(5e+5)-1)/(p^5-1). - _David W. Wilson_, Aug 01 2001
%F G.f.: sum(k>=1, k^5*x^k/(1-x^k)). - _Benoit Cloitre_, Apr 21 2003
%F Dirichlet g.f.: zeta(s)*zeta(s-5). - _R. J. Mathar_, Mar 06 2011
%F G.f. also (1 - E_6(q))/540, with the g.f. E_6 of A013973. See Hardy p. 166, (10.5.7) with R = E_6. - _Wolfdieter Lang_, Jan 31 2017
%F L.g.f.: -log(Product_{k>=1} (1 - x^k)^(k^4)) = Sum_{n>=1} a(n)*x^n/n. - _Ilya Gutkovskiy_, May 06 2017
%F a(n) = Sum_{1 <= i, j, k, l, m <= n} tau(gcd(i, j, k, l, m, n)) = Sum_{d divides n} tau(d) * J_5(n/d), where the divisor function tau(n) = A000005(n) and the Jordan totient function J_5(n) = A059378(n). - _Peter Bala_, Jan 22 2024
%p A001160 := proc(n)
%p numtheory[sigma][5](n);
%p end proc:
%p seq(A001160(n),n=1..30) ; # _R. J. Mathar_, Jan 31 2017
%t lst={};Do[AppendTo[lst,DivisorSigma[5,n]],{n,5!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Mar 11 2009 *)
%t DivisorSigma[5,Range[30]] (* _Harvey P. Dale_, Nov 11 2013 *)
%o (Sage) [sigma(n, 5) for n in range(1, 30)] # _Zerinvary Lajos_, Jun 04 2009
%o (PARI) a(n)=sigma(n,5) \\ _Charles R Greathouse IV_, Apr 28 2011
%o (Magma) [DivisorSigma(5,n): n in [1..30]]; // _Bruno Berselli_, Apr 10 2013
%Y Cf. A000005, A000203, A001157, A001158, A001159, A013973, A000584 (Mobius transform), A178448 (Dirichlet inverse)
%K nonn,easy,mult
%O 1,2
%A _N. J. A. Sloane_
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