A318742 a(n) = Sum_{k=1..n} floor(n/k)^3.

1, 9, 29, 74, 136, 254, 382, 596, 833, 1173, 1505, 2057, 2527, 3209, 3921, 4856, 5674, 6928, 7956, 9474, 10882, 12608, 14128, 16506, 18369, 20797, 23141, 26129, 28567, 32259, 35051, 38963, 42483, 46675, 50435, 55904, 59902, 65156, 70092, 76460

Offset: 1

Author

Vaclav Kotesovec, Sep 02 2018

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000
• Richard J. Mathar, Chebyshev approximation of x^m (-log x)^l in the interval 0 <= x <= 1, arXiv:2408.15212 [math.CA], 2024. See p. 2.

Crossrefs

Cf. A000005, A000203, A024916, A064602, A064603, A064604.

Cf. A006218, A222548, A318743, A318744.

Programs

Magma

[&+[Floor(n/k)^3:k in [1..n]]: n in [1..40]]; // Marius A. Burtea, Jul 16 2019

Mathematica

Table[Sum[Floor[n/k]^3, {k, 1, n}], {n, 1, 40}]
Accumulate[Table[DivisorSigma[0, k] - 3*DivisorSigma[1, k] + 3*DivisorSigma[2, k], {k, 1, 40}]]

PARI

a(n) = sum(k=1, n, (n\k)^3); \\ Michel Marcus, Sep 03 2018

Python

from math import isqrt
def A318742(n): return -(s:=isqrt(n))**4 + sum((q:=n//k)*(3*k*(k-1)+q**2+1) for k in range(1,s+1)) # Chai Wah Wu, Oct 21 2023

Formula

a(n) = A006218(n) - 3*A024916(n) + 3*A064602(n).

a(n) ~ zeta(3) * n^3.

G.f.: (1/(1 - x)) * Sum_{k>=1} (3*k*(k - 1) + 1) * x^k/(1 - x^k). - Ilya Gutkovskiy, Jul 16 2019