A319194 - OEIS

%I #22 Jul 16 2024 04:19:19

%S 1,6,38,373,4461,68033,1202753,24757484,574608039,14925278329,

%T 427729375161,13424413453317,457608305315211,16841852554413561,

%U 665483754539870667,28101844918556128030,1262901795439193700478,60182608193322255156347,3031285556584399354961535

%N a(n) = Sum_{k=1..n} sigma(n,k).

%H Vaclav Kotesovec, <a href="/A319194/b319194.txt">Table of n, a(n) for n = 1..380</a>

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

%F a(n) = Sum_{k=1..n} k^n * floor(n/k). - _Daniel Suteu_, Nov 10 2018

%p with(NumberTheory): seq(sum(sigma[n](k), k = 1..n), n = 1..20); # _Vaclav Kotesovec_, Aug 20 2019

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

%o (PARI) a(n) = sum(k=1, n, sigma(k,n)); \\ _Michel Marcus_, Sep 13 2018

%o (PARI) a(n) = sum(k=1, n, k^n * (n\k)); \\ _Daniel Suteu_, Nov 10 2018

%o (Python)

%o from math import isqrt

%o from sympy import bernoulli

%o 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

%Y Cf. A065805, A308652.

%K nonn

%O 1,2

%A _Vaclav Kotesovec_, Sep 13 2018