A280375 Expansion of Sum_{k>=1} k^3*x^(k^2)/(1 - x^k).

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

Offset

1,4

Comments

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

Links

Table of n, a(n) for n=1..85.

Index entries for sequences related to sums of divisors

Formula

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

Example

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

Mathematica

nmax = 85; Rest[CoefficientList[Series[Sum[k^3 x^k^2/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
(* Second program *)
Table[Total[Select[Divisors@ n, # <= Sqrt@ n &]^3], {n, 85}] (* Michael De Vlieger, Jan 01 2017 *)

Prog

(PARI) a(n) = my(rn = sqrt(n)); sumdiv(n, d, d^3*(d<=rn)); \\ Michel Marcus, Jan 02 2017

Crossrefs

Cf. A001158, A038548, A066839, A095118.

Sequence in context: A224835 A329725 A339354 * A067617 A230458 A246687

Adjacent sequences: A280372 A280373 A280374 * A280376 A280377 A280378

Keyword

nonn

Author

Ilya Gutkovskiy, Jan 01 2017

Status

approved