A332469 a(n) = Sum_{k=1..n} floor(n/k)^n.

1, 5, 29, 274, 3160, 47452, 825862, 16843268, 387702833, 10009826727, 285360679985, 8918294547447, 302888236005847, 11112685321898449, 437898668488710801, 18447025705612363530, 827242514466399305122, 39346558271561286347116, 1978421007121668206129316

Offset: 1

Author

Ilya Gutkovskiy, Feb 13 2020

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..386

Crossrefs

Cf. A006218, A222548, A318742, A318743, A318744, A319194, A332468.

Programs

Magma

[&+[Floor(n/k)^n:k in [1..n]]:n in [1..20]]; // Marius A. Burtea, Feb 13 2020

Mathematica

Table[Sum[Floor[n/k]^n, {k, 1, n}], {n, 1, 19}]
Table[SeriesCoefficient[1/(1 - x) Sum[(k^n - (k - 1)^n) x^k/(1 - x^k), {k, 1, n}], {x, 0, n}], {n, 1, 19}]

PARI

a(n)={sum(k=1, n, floor(n/k)^n)} \\ Andrew Howroyd, Feb 13 2020

Python

from math import isqrt
def A332469(n): return -(s:=isqrt(n))**(n+1)+sum((q:=n//k)*(k**n-(k-1)**n+q**(n-1)) for k in range(1,s+1)) # Chai Wah Wu, Oct 26 2023

Formula

a(n) = [x^n] (1/(1 - x)) * Sum_{k>=1} (k^n - (k - 1)^n) * x^k / (1 - x^k).

a(n) ~ n^n. - Vaclav Kotesovec, Jun 11 2021