A356130 - OEIS

%I #20 Oct 21 2023 19:39:06

%S 1,4,16,111,999,12513,185683,3316418,67810767,1576561677,40862702931,

%T 1171104916405,36722498575799,1251419967587955,46034784688102781,

%U 1818440444592581068,76763036794222996512,3448830049286378614987,164309958491233496689189

%N a(n) = Sum_{k=1..n} sigma_{n-1}(k).

%H Seiichi Manyama, <a href="/A356130/b356130.txt">Table of n, a(n) for n = 1..387</a>

%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>

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

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

%t a[n_] := Sum[DivisorSigma[n-1, k], {k, 1, n}]; Array[a, 19] (* _Amiram Eldar_, Jul 28 2022 *)

%o (PARI) a(n) = sum(k=1, n, sigma(k, n-1));

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

%o (Python)

%o from math import isqrt

%o from sympy import bernoulli

%o def A350130(n): return (((s:=isqrt(n))+1)*((b:=bernoulli(n))-bernoulli(n, s+1))+sum(k**(n-1)*n*((q:=n//k)+1)-b+bernoulli(n, q+1) for k in range(1,s+1)))//n if n>1 else 1 # _Chai Wah Wu_, Oct 21 2023

%Y Cf. A319194, A319649, A356129.

%K nonn

%O 1,2

%A _Seiichi Manyama_, Jul 27 2022