The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A051002 Sum of 5th powers of odd divisors of n. 14
 1, 1, 244, 1, 3126, 244, 16808, 1, 59293, 3126, 161052, 244, 371294, 16808, 762744, 1, 1419858, 59293, 2476100, 3126, 4101152, 161052, 6436344, 244, 9768751, 371294, 14408200, 16808, 20511150, 762744, 28629152, 1, 39296688, 1419858, 52541808, 59293, 69343958 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS The Apostol exercise F(x) is the g.f. of a(n)*(-1)^(n+1). - Michael Somos, Jul 05 2021 REFERENCES T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer-Verlag, 1990, page 25, Exercise 15. LINKS Seiichi Manyama, Table of n, a(n) for n = 1..10000 J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8). Eric Weisstein's World of Mathematics, Odd Divisor Function. FORMULA Dirichlet g.f.: (1-2^(5-s))*zeta(s)*zeta(s-5). - R. J. Mathar, Apr 06 2011 G.f.: Sum_{k>=1} (2*k - 1)^5*x^(2*k-1)/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Jan 04 2017 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 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 Sum_{k=1..n} a(k) ~ Pi^6 * n^6 / 11340. - Vaclav Kotesovec, Sep 24 2020 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 EXAMPLE 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 MATHEMATICA a[n_] := Select[ Divisors[n], OddQ]^5 // Total; Table[a[n], {n, 1, 34}] (* Jean-François Alcover, Oct 25 2012 *) 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 *) a[ n_] := If[n == 0, 0, DivisorSigma[5, n / 2^IntegerExponent[n, 2]]]; (* Michael Somos, Jul 05 2021 *) PROG (PARI) a(n) = sumdiv(n , d, (d%2)*d^5); \\ Michel Marcus, Jan 14 2014 (PARI) a(n)=sumdiv(n>>valuation(n, 2), d, d^5) \\ Charles R Greathouse IV, Jul 05 2021 (Python) from sympy import divisor_sigma def A051002(n): return int(divisor_sigma(n>>(~n&n-1).bit_length(), 5)) # Chai Wah Wu, Jul 16 2022 CROSSREFS Cf. A000593, A001227, A050999, A051000, A051001, A001160. Sequence in context: A352033 A248137 A243774 * A044987 A201998 A234262 Adjacent sequences:  A050999 A051000 A051001 * A051003 A051004 A051005 KEYWORD nonn,mult AUTHOR STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified August 11 05:13 EDT 2022. Contains 356046 sequences. (Running on oeis4.)