A356239 a(n) = Sum_{k=1..n} k^n * sigma_0(k).

1, 9, 71, 963, 9873, 231749, 2976863, 86348423, 1824883450, 55584932826, 1104642697680, 64932555347084, 1366828157222090, 61273696016238014, 2581786206601959958, 129797968403021602450, 3678372903755436314440, 295835829367866540495396

Offset

1,2

Links

Robert Israel, Table of n, a(n) for n = 1..384

Formula

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

Maple

f:= proc(n) local k; add(k^n * numtheory:-tau(k), k=1..n) end proc:
map(f, [$1..30]); # Robert Israel, Jan 21 2024

Mathematica

a[n_] := Sum[k^n * DivisorSigma[0, k], {k, 1, n}]; Array[a, 18] (* Amiram Eldar, Jul 30 2022 *)

Prog

(PARI) a(n) = sum(k=1, n, k^n*sigma(k, 0));
(PARI) a(n) = sum(k=1, n, k^n*sum(j=1, n\k, j^n));
(Python)
from math import isqrt
from sympy import bernoulli
def A356239(n): return (-(bernoulli(n+1, (s:=isqrt(n))+1)-(b:=bernoulli(n+1)))**2//(n+1) + sum(k**n*(bernoulli(n+1, n//k+1)-b)<<1 for k in range(1, s+1)))//(n+1) # Chai Wah Wu, Oct 21 2023

Crossrefs

Cf. A000005, A319194, A356129, A356243.

Sequence in context: A001706 A368268 A251284 * A324413 A144745 A158193

Adjacent sequences: A356236 A356237 A356238 * A356240 A356241 A356242

Keyword

nonn

Author

Seiichi Manyama, Jul 30 2022

Status

approved