A350128 a(n) = Sum_{k=1..n} k^n * floor(n/k)^2.

1, 8, 44, 417, 4545, 69905, 1207937, 24904806, 575256641, 14947281595, 427836523971, 13429362462839, 457637290140469, 16843379604615375, 665494379869134005, 28102480944522059434, 1262906802939553227382, 60182948301301262753877

Offset: 1

Author

Seiichi Manyama, Dec 15 2021

Keywords

• nonn

Links

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

Crossrefs

Cf. A222548, A319194, A350107, A350125.

Programs

Mathematica

Table[Sum[k^n Floor[n/k]^2,{k,n}],{n,20}] (* Harvey P. Dale, Feb 11 2022 *)

PARI

a(n) = sum(k=1, n, k^n*(n\k)^2);

PARI

a(n) = sum(k=1, n, 2*k*sigma(k, n-1)-sigma(k, n));

Python

from math import isqrt
from sympy import bernoulli
def A350128(n): return (((s:=isqrt(n))+1)*(1-s)*(bernoulli(n+1,s+1)-(b:=bernoulli(n+1)))+sum(k**n*(n+1)*(((q:=n//k)+1)*(q-1))+(1-2*k)*(b-bernoulli(n+1,q+1)) for k in range(1,s+1)))//(n+1) # Chai Wah Wu, Oct 21 2023

Formula

a(n) = Sum_{k=1..n} 2 * k * sigma_{n-1}(k) - sigma_{n}(k).

a(n) ~ n^n / (1 - exp(-1)). - Vaclav Kotesovec, Dec 16 2021