A285989 a(0) = 0, a(n) = Sum_{0 0.

0, 1, 16, 82, 256, 626, 1312, 2402, 4096, 6643, 10016, 14642, 20992, 28562, 38432, 51332, 65536, 83522, 106288, 130322, 160256, 196964, 234272, 279842, 335872, 391251, 456992, 538084, 614912, 707282, 821312, 923522, 1048576, 1200644, 1336352, 1503652, 1700608

Offset: 0

Author

Seiichi Manyama, Apr 30 2017

Keywords

• nonn
• mult

Comments

Multiplicative because this sequence is the Dirichlet convolution of A000035 and A000583 which are both multiplicative. - Andrew Howroyd, Aug 05 2018

Links

• Robert Israel, Table of n, a(n) for n = 0..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).
• Index entries for sequences mentioned by Glaisher.

Crossrefs

Sum_{0A002131 (k=1), A076577 (k=2), A007331 (k=3), this sequence (k=4), A096960 (k=5), A096961 (k=7), A096962 (k=9), A096963 (k=11).

Cf. A000035, A000583, A007814, A013663, A051001, A206623, A285990.

Programs

Maple

f:= n -> add((n/d)^4, d = numtheory:-divisors(n/2^padic:-ordp(n,2))); # Robert Israel, Apr 30 2017

Mathematica

{0}~Join~Table[DivisorSum[n, Mod[#, 2] (n/#)^4 &], {n, 36}] (* Michael De Vlieger, Aug 05 2018 *)

PARI

a(n)={sumdiv(n, d, (d%2)*(n/d)^4)} \\ Andrew Howroyd, Aug 05 2018

Formula

a(n) = A051001(n)*16^A007814(n) for n >= 1. - Robert Israel, Apr 30 2017

From Amiram Eldar, Nov 01 2022: (Start)

Multiplicative with a(2^e) = 2^(4*e) and a(p^e) = (p^(4*e+4)-1)/(p^4-1) for p > 2.

Sum_{k=1..n} a(k) ~ c * n^5, where c = 31*zeta(5)/160 = 0.200904... . (End)

Dirichlet g.f.: zeta(s)*zeta(s-4)*(1-1/2^s). - Amiram Eldar, Jan 08 2023