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%I #49 Jul 16 2022 17:52:50
%S 1,1,244,1,3126,244,16808,1,59293,3126,161052,244,371294,16808,762744,
%T 1,1419858,59293,2476100,3126,4101152,161052,6436344,244,9768751,
%U 371294,14408200,16808,20511150,762744,28629152,1,39296688,1419858,52541808,59293,69343958
%N Sum of 5th powers of odd divisors of n.
%C The Apostol exercise F(x) is the g.f. of a(n)*(-1)^(n+1). - _Michael Somos_, Jul 05 2021
%D T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer-Verlag, 1990, page 25, Exercise 15.
%H Seiichi Manyama, <a href="/A051002/b051002.txt">Table of n, a(n) for n = 1..10000</a>
%H J. W. L. Glaisher, <a href="https://books.google.com/books?id=bLs9AQAAMAAJ&pg=RA1-PA1">On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares</a>, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/OddDivisorFunction.html">Odd Divisor Function</a>.
%H <a href="/index/Ge#Glaisher">Index entries for sequences mentioned by Glaisher</a>
%F Dirichlet g.f.: (1-2^(5-s))*zeta(s)*zeta(s-5). - _R. J. Mathar_, Apr 06 2011
%F G.f.: Sum_{k>=1} (2*k - 1)^5*x^(2*k-1)/(1 - x^(2*k-1)). - _Ilya Gutkovskiy_, Jan 04 2017
%F The preceding g.f. is also 34*sigma_5(x^2) - 64*sigma_5(x^4) - sigma_5(-x), with sigma_5 the g.f. of A001160. Compare this with the Apostol reference which gives the g.f. of a(n)*(-1)^(n+1). - _Wolfdieter Lang_, Jan 31 2017
%F Multiplicative with a(2^e) = 1 and a(p^e) = (p^(5*e+5)-1)/(p^5-1) for p > 2. - _Amiram Eldar_, Sep 14 2020
%F Sum_{k=1..n} a(k) ~ Pi^6 * n^6 / 11340. - _Vaclav Kotesovec_, Sep 24 2020
%F G.f.: Sum_{n >= 1} x^n*R(5,x^(2*n))/(1 - x^(2*n))^6, where R(5,x) = 1 + 237*x + 1682*x^2 + 1682*x^3 + 237*x^4 + x^5 is the fifth row polynomial of A060187. - _Peter Bala_, Dec 20 2021
%e G.f. = x + x^2 + 244*x^3 + x^4 + 3126*x^5 + 244*x^6 + 16808*x^7 + x^8 + ... - _Michael Somos_, Jul 05 2021
%t a[n_] := Select[ Divisors[n], OddQ]^5 // Total; Table[a[n], {n, 1, 34}] (* _Jean-François Alcover_, Oct 25 2012 *)
%t f[2, e_] := 1; f[p_, e_] := (p^(5*e + 5) - 1)/(p^5 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Sep 14 2020 *)
%t a[ n_] := If[n == 0, 0, DivisorSigma[5, n / 2^IntegerExponent[n, 2]]]; (* _Michael Somos_, Jul 05 2021 *)
%o (PARI) a(n) = sumdiv(n , d, (d%2)*d^5); \\ _Michel Marcus_, Jan 14 2014
%o (PARI) a(n)=sumdiv(n>>valuation(n,2), d, d^5) \\ _Charles R Greathouse IV_, Jul 05 2021
%o (Python)
%o from sympy import divisor_sigma
%o def A051002(n): return int(divisor_sigma(n>>(~n&n-1).bit_length(),5)) # _Chai Wah Wu_, Jul 16 2022
%Y Cf. A000593, A001227, A050999, A051000, A051001, A001160.
%K nonn,mult
%O 1,3
%A _Eric W. Weisstein_
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