|
|
A051002
|
|
Sum of 5th powers of odd divisors of n.
|
|
16
|
|
|
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
|
|
|
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
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
(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
|
|
|
KEYWORD
|
nonn,mult
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|