OFFSET
1,3
LINKS
Robert Israel, 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^(4-s))*zeta(s)*zeta(s-4). - R. J. Mathar, Apr 06 2011
G.f.: Sum_{k>=1} (2*k - 1)^4*x^(2*k-1)/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Jan 04 2017
Multiplicative with a(2^e) = 1 and a(p^e) = (p^(4*e+4)-1)/(p^4-1) for p > 2. - Amiram Eldar, Sep 14 2020
Sum_{k=1..n} a(k) ~ zeta(5)*n^5/10. - Vaclav Kotesovec, Sep 24 2020
G.f.: Sum_{n >= 1} x^n*(1 + 76*x^(2*n) + 230*x^(4*n) + 76*x^(6*n) + x^(8*n))/(1 - x^(2*n))^5. See row 5 of A060187. - Peter Bala, Dec 20 2021
MAPLE
f:= proc(n) add(x^4, x = numtheory:-divisors(n/2^padic:-ordp(n, 2))) end proc:
map(f, [$1..100]); # Robert Israel, Jan 05 2017
MATHEMATICA
Table[Total[Select[Divisors[n], OddQ]^4], {n, 40}] (* Harvey P. Dale, Oct 02 2014 *)
f[2, e_] := 1; f[p_, e_] := (p^(4*e + 4) - 1)/(p^4 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 14 2020 *)
PROG
(PARI) a(n) = sumdiv(n , d, (d%2)*d^4); \\ Michel Marcus, Jan 14 2014
(Python)
from sympy import divisor_sigma
def A051001(n): return int(divisor_sigma(n>>(~n&n-1).bit_length(), 4)) # Chai Wah Wu, Jul 16 2022
CROSSREFS
KEYWORD
AUTHOR
STATUS
approved