login
a(n) = Sum_{0<d|n, n/d odd} d^5.
10

%I #45 Jan 09 2023 01:45:37

%S 1,32,244,1024,3126,7808,16808,32768,59293,100032,161052,249856,

%T 371294,537856,762744,1048576,1419858,1897376,2476100,3201024,4101152,

%U 5153664,6436344,7995392,9768751,11881408,14408200,17211392,20511150

%N a(n) = Sum_{0<d|n, n/d odd} d^5.

%H Seiichi Manyama, <a href="/A096960/b096960.txt">Table of n, a(n) for n = 1..10000</a>

%H H. H. Chan and C. Krattenthaler, <a href="https://arxiv.org/abs/math/0407061">Recent progress in the study of representations of integers as sums of squares</a>, arXiv:math/0407061 [math.NT], 2004.

%H J. W. L. Glaisher, <a href="https://books.google.com/books?id=bLs9AQAAMAAJ&amp;pg=RA1-PA1">On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares</a>, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).

%H <a href="/index/Ge#Glaisher">Index entries for sequences mentioned by Glaisher</a>.

%F G.f.: Sum {k>0} k^5 * x^k / (1 - x^(2*k)).

%F From _Amiram Eldar_, Nov 01 2022: (Start)

%F Multiplicative with a(2^e) = 2^(5*e) and a(p^e) = (p^(5*e+5)-1)/(p^5-1) for p > 2.

%F Sum_{k=1..n} a(k) ~ c * n^6, where c = 21*zeta(6)/128 = 0.166907... . (End)

%F Dirichlet g.f.: zeta(s)*zeta(s-5)*(1-1/2^s). - _Amiram Eldar_, Jan 08 2023

%e G.f. = q + 32*q^2 + 244*q^3 + 1024*q^4 + 3126*q^5 + 7808*q^6 + 16808*q^7 + 32768*q^8 + ...

%t a[ n_] := If[ n < 1, 0, Sum[ d^5 Boole[ OddQ[ n/d]], {d, Divisors[ n]}]]; (* _Michael Somos_, Jun 04 2013 *)

%t a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q]^4 + EllipticTheta[ 2, 0, q]^4) EllipticTheta[ 2, 0, q^(1/2)]^8 / 256, {q, 0, n}]; (* _Michael Somos_, Jun 04 2013 *)

%o (PARI) {a(n) = if( n<1, 0, sumdiv( n, d, (n/d%2) * d^5))};

%o (Sage)

%o ModularForms( Gamma0(2), 6, prec=33).gen(1).coefficients(30) # _Michael Somos_, Jun 04 2013

%o (Magma) Basis( ModularForms( Gamma0(2), 6), 30) [2]; /* _Michael Somos_, Nov 30 2014 */

%Y Cf. A007331, A013664, A096961, A096962, A096963.

%K nonn,mult

%O 1,2

%A _Ralf Stephan_, Jul 18 2004