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

%I A356239 #25 Jan 21 2024 18:10:13
%S A356239 1,9,71,963,9873,231749,2976863,86348423,1824883450,55584932826,
%T A356239 1104642697680,64932555347084,1366828157222090,61273696016238014,
%U A356239 2581786206601959958,129797968403021602450,3678372903755436314440,295835829367866540495396
%N A356239 a(n) = Sum_{k=1..n} k^n * sigma_0(k).
%H A356239 Robert Israel, <a href="/A356239/b356239.txt">Table of n, a(n) for n = 1..384</a>
%F A356239 a(n) = Sum_{k=1..n} k^n * Sum_{j=1..floor(n/k)} j^n.
%p A356239 f:= proc(n) local k; add(k^n * numtheory:-tau(k),k=1..n) end proc:
%p A356239 map(f, [$1..30]); # _Robert Israel_, Jan 21 2024
%t A356239 a[n_] := Sum[k^n * DivisorSigma[0, k], {k, 1, n}]; Array[a, 18] (* _Amiram Eldar_, Jul 30 2022 *)
%o A356239 (PARI) a(n) = sum(k=1, n, k^n*sigma(k, 0));
%o A356239 (PARI) a(n) = sum(k=1, n, k^n*sum(j=1, n\k, j^n));
%o A356239 (Python)
%o A356239 from math import isqrt
%o A356239 from sympy import bernoulli
%o A356239 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
%Y A356239 Cf. A000005, A319194, A356129, A356243.
%K A356239 nonn
%O A356239 1,2
%A A356239 _Seiichi Manyama_, Jul 30 2022