A356100 a(n) = Sum_{k=1..n} (k - 1)^n * floor(n/k).

0, 1, 9, 99, 1301, 20581, 376891, 7914216, 186905206, 4915451602, 142368695176, 4506118905870, 154720069309364, 5729167232515112, 227585086051159866, 9654819212943764500, 435659280972794395356, 20836049921760968809231, 1052864549462731148832219

Offset: 1

Author

Seiichi Manyama, Jul 26 2022

Keywords

• nonn

Crossrefs

Cf. A121706, A236632, A319194, A332469.

Programs

Mathematica

Table[Sum[(k-1)^n Floor[n/k],{k,n}],{n,20}] (* Harvey P. Dale, Dec 14 2024 *)

PARI

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

PARI

a(n) = sum(k=1, n, sigma(k, n)-(n\k)^n);

PARI

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

Python

def A356100(n): return sum((k-1)**n*(n//k) for k in range(2,n+1)) # Chai Wah Wu, Jul 26 2022

Formula

a(n) = A319194(n) - A332469(n).

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

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