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

%I A319194 #26 Mar 02 2025 07:59:44
%S A319194 1,6,38,373,4461,68033,1202753,24757484,574608039,14925278329,
%T A319194 427729375161,13424413453317,457608305315211,16841852554413561,
%U A319194 665483754539870667,28101844918556128030,1262901795439193700478,60182608193322255156347,3031285556584399354961535
%N A319194 a(n) = Sum_{k=1..n} sigma_n(k).
%H A319194 Vaclav Kotesovec, <a href="/A319194/b319194.txt">Table of n, a(n) for n = 1..380</a>
%F A319194 a(n) ~ n^n / (1 - exp(-1)).
%F A319194 a(n) = Sum_{k=1..n} k^n * floor(n/k). - _Daniel Suteu_, Nov 10 2018
%p A319194 with(NumberTheory): seq(sum(sigma[n](k), k = 1..n), n = 1..20); # _Vaclav Kotesovec_, Aug 20 2019
%t A319194 Table[Sum[DivisorSigma[n, k], {k, 1, n}], {n, 1, 20}]
%o A319194 (PARI) a(n) = sum(k=1, n, sigma(k,n)); \\ _Michel Marcus_, Sep 13 2018
%o A319194 (PARI) a(n) = sum(k=1, n, k^n * (n\k)); \\ _Daniel Suteu_, Nov 10 2018
%o A319194 (Python)
%o A319194 from math import isqrt
%o A319194 from sympy import bernoulli
%o A319194 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 A319194 Cf. A065805, A308652.
%K A319194 nonn
%O A319194 1,2
%A A319194 _Vaclav Kotesovec_, Sep 13 2018