A319194 a(n) = Sum_{k=1..n} sigma_n(k).

1, 6, 38, 373, 4461, 68033, 1202753, 24757484, 574608039, 14925278329, 427729375161, 13424413453317, 457608305315211, 16841852554413561, 665483754539870667, 28101844918556128030, 1262901795439193700478, 60182608193322255156347, 3031285556584399354961535

Offset: 1

Author

Vaclav Kotesovec, Sep 13 2018

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..380

Crossrefs

Cf. A065805, A308652.

Programs

Maple

with(NumberTheory): seq(sum(sigma[n](k), k = 1..n), n = 1..20); # Vaclav Kotesovec, Aug 20 2019

Mathematica

Table[Sum[DivisorSigma[n, k], {k, 1, n}], {n, 1, 20}]

PARI

a(n) = sum(k=1, n, sigma(k,n)); \\ Michel Marcus, Sep 13 2018

PARI

a(n) = sum(k=1, n, k^n * (n\k)); \\ Daniel Suteu, Nov 10 2018

Python

from math import isqrt
from sympy import bernoulli
def A319194(n): return (((s:=isqrt(n))+1)*((b:=bernoulli(n+1))-bernoulli(n+1,s+1))+sum(k**n*(n+1)*((q:=n//k)+1)-b+bernoulli(n+1,q+1) for k in range(1,s+1)))//(n+1) # Chai Wah Wu, Oct 21 2023

Formula

a(n) ~ n^n / (1 - exp(-1)).

a(n) = Sum_{k=1..n} k^n * floor(n/k). - Daniel Suteu, Nov 10 2018