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Expansion of Sum_{k>=1} k^3*x^(k^2)/(1 - x^k).
4

%I #15 Jan 02 2017 20:34:41

%S 1,1,1,9,1,9,1,9,28,9,1,36,1,9,28,73,1,36,1,73,28,9,1,100,126,9,28,73,

%T 1,161,1,73,28,9,126,316,1,9,28,198,1,252,1,73,153,9,1,316,344,134,28,

%U 73,1,252,126,416,28,9,1,441,1,9,371,585,126,252,1,73,28,477,1,828,1,9,153,73,344,252,1,710,757,9,1,659,126

%N Expansion of Sum_{k>=1} k^3*x^(k^2)/(1 - x^k).

%C The sum of the cubes of the divisors of n which are <= sqrt(n).

%H <a href="/index/Su#sums_of_divisors">Index entries for sequences related to sums of divisors</a>

%F G.f.: Sum_{k>=1} k^3*x^(k^2)/(1 - x^k).

%e The divisors of 12 which are <= sqrt(12) are {1,2,3}, so a(12) = 1^3 + 2^3 + 3^3 = 36.

%t nmax = 85; Rest[CoefficientList[Series[Sum[k^3 x^k^2/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]]

%t (* Second program *)

%t Table[Total[Select[Divisors@ n, # <= Sqrt@ n &]^3], {n, 85}] (* _Michael De Vlieger_, Jan 01 2017 *)

%o (PARI) a(n) = my(rn = sqrt(n)); sumdiv(n, d, d^3*(d<=rn)); \\ _Michel Marcus_, Jan 02 2017

%Y Cf. A001158, A038548, A066839, A095118.

%K nonn

%O 1,4

%A _Ilya Gutkovskiy_, Jan 01 2017