A344675 a(n) = Sum_{k=1..n} floor(n^3/k^3).

1, 9, 31, 75, 146, 256, 406, 610, 870, 1194, 1590, 2069, 2631, 3286, 4043, 4910, 5889, 6997, 8228, 9600, 11114, 12781, 14605, 16599, 18760, 21106, 23636, 26363, 29292, 32429, 35781, 39359, 43169, 47212, 51505, 56054, 60855, 65924, 71268, 76898, 82807, 89021

Offset

1,2

Comments

In general, for m > 1, Sum_{k=1..n} floor(n^m/k^m) ~ zeta(m)*n^m + zeta(1/m)*n.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Vaclav Kotesovec, Plot of (a(n) - zeta(3)*n^3)/n for n = 1..100000

Formula

a(n) ~ zeta(3)*n^3 + zeta(1/3)*n.

Mathematica

Table[Sum[Floor[n^3/k^3], {k, 1, n}], {n, 1, 50}]

Prog

(Python)
from sympy import integer_nthroot
def A344675(n):
    c, j, n2 = 0, 1, n**3
    while j <= n:
        k = n2//j**3
        m = integer_nthroot(n2//k, 3)[0]
        c += k*(m-j+1)
        j = m+1
    return c # Chai Wah Wu, May 21 2026

Crossrefs

Cf. A001158, A006218, A153818.

Sequence in context: A288419 A168297 A004126 * A177342 A224000 A373061

Adjacent sequences: A344672 A344673 A344674 * A344676 A344677 A344678

Keyword

nonn

Author

Vaclav Kotesovec, May 26 2021

Status

approved