A356249 a(n) = Sum_{k=1..n} (k * floor(n/k))^3.

1, 16, 62, 219, 405, 1053, 1523, 2948, 4407, 7041, 8703, 15283, 17949, 24657, 32685, 44806, 50536, 70687, 78573, 105411, 125879, 149879, 163565, 222425, 247476, 286134, 327634, 396258, 423084, 532236, 564818, 664763, 738095, 821693, 904937, 1107618, 1162268, 1277588, 1395760

Offset: 1

Author

Seiichi Manyama, Jul 31 2022

Keywords

• nonn

Links

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

Crossrefs

Column k=3 of A356250.

Cf. A000537, A024916, A318742, A350123.

Cf. A064603, A356125, A319086.

Programs

Mathematica

a[n_] := Sum[(k * Floor[n/k])^3, {k, 1, n}]; Array[a, 40] (* Amiram Eldar, Jul 31 2022 *)

PARI

a(n) = sum(k=1, n, (k*(n\k))^3);

PARI

a(n) = sum(k=1, n, k^3*sumdiv(k, d, 1-(1-1/d)^3));

PARI

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (k^3-(k-1)^3)*x^k*(1+4*x^k+x^(2*k))/(1-x^k)^4)/(1-x))

Python

from math import isqrt
def A356249(n): return -(s:=isqrt(n))**5*(s+1)**2 + sum((q:=n//k)**2*(k*(3*(k-1))+q*(k*(k*(4*k+6)-6)+q*(k*(3*(k-1))+1)+2)+1) for k in range(1,s+1))>>2 # Chai Wah Wu, Oct 21 2023

Formula

a(n) = Sum_{k=1..n} k^3 * Sum_{d|k} (1 - (1 - 1/d)^3).

G.f.: (1/(1 - x)) * Sum_{k>=1} (k^3 - (k - 1)^3) * x^k * (1 + 4*x^k + x^(2*k))/(1 - x^k)^4.

From Vaclav Kotesovec, Aug 02 2022: (Start)

a(n) = A064603(n) - 3*A356125(n) + 3*A319086(n).

a(n) ~ n^4 * (Pi^2/8 + Pi^4/360 - 3*zeta(3)/4). (End)