A309125 a(n) = n + 2^2 * floor(n/2^2) + 3^2 * floor(n/3^2) + 4^2 * floor(n/4^2) + ...

1, 2, 3, 8, 9, 10, 11, 16, 26, 27, 28, 33, 34, 35, 36, 57, 58, 68, 69, 74, 75, 76, 77, 82, 108, 109, 119, 124, 125, 126, 127, 148, 149, 150, 151, 201, 202, 203, 204, 209, 210, 211, 212, 217, 227, 228, 229, 250, 300, 326, 327, 332, 333, 343, 344, 349, 350, 351, 352, 357, 358, 359, 369, 454, 455, 456

Offset

1,2

Comments

Partial sums of A035316.

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Vaclav Kotesovec, Plot of a(n)/n^(3/2) for n = 1..10000

Formula

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

a(n) ~ zeta(3/2)*n^(3/2)/3 - n/2. - Vaclav Kotesovec, Aug 30 2021

Mathematica

Table[Sum[k^2 Floor[n/k^2], {k, 1, n}], {n, 1, 66}]
nmax = 66; CoefficientList[Series[1/(1 - x) Sum[k^2 x^(k^2)/(1 - x^(k^2)), {k, 1, Floor[nmax^(1/2)] + 1}], {x, 0, nmax}], x] // Rest

Prog

(PARI) a(n) = sum(k=1, n, k^2*(n\k^2)); \\ Seiichi Manyama, Aug 30 2021

Crossrefs

Cf. A013936, A024916, A035316, A309126, A309127.

Sequence in context: A047476 A347397 A037462 * A037318 A253317 A277971

Adjacent sequences: A309122 A309123 A309124 * A309126 A309127 A309128

Keyword

nonn

Author

Ilya Gutkovskiy, Jul 13 2019

Status

approved