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
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
Programs
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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
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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 *) -
PARI
a(n) = sumdiv(n , d, (d%2)*d^4); \\ Michel Marcus, Jan 14 2014
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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
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