A051001 Sum of 4th powers of odd divisors of n.

1, 1, 82, 1, 626, 82, 2402, 1, 6643, 626, 14642, 82, 28562, 2402, 51332, 1, 83522, 6643, 130322, 626, 196964, 14642, 279842, 82, 391251, 28562, 538084, 2402, 707282, 51332, 923522, 1, 1200644, 83522, 1503652, 6643, 1874162, 130322, 2342084, 626, 2825762, 196964

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.

Index entries for sequences mentioned by Glaisher

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

a(n) = A001159(A000265(n)). - Robert Israel, Jan 05 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

Cf. A000265, A000593, A001227, A001159, A050999, A051000, A051002.

Sequence in context: A347172 A347175 A352032 * A363991 A050678 A354501

Adjacent sequences: A050998 A050999 A051000 * A051002 A051003 A051004

Keyword

nonn,mult,look

Author

Eric W. Weisstein

Status

approved