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A385542 The sum of the aliquot divisors of n that are powerful.

%I A385542 #11 Jul 05 2025 08:59:32
%S A385542 0,1,1,1,1,1,1,5,1,1,1,5,1,1,1,13,1,10,1,5,1,1,1,13,1,1,10,5,1,1,1,29,
%T A385542 1,1,1,14,1,1,1,13,1,1,1,5,10,1,1,29,1,26,1,5,1,37,1,13,1,1,1,5,1,1,
%U A385542 10,61,1,1,1,5,1,1,1,58,1,1,26,5,1,1,1,29,37
%N A385542 The sum of the aliquot divisors of n that are powerful.
%H A385542 Amiram Eldar, <a href="/A385542/b385542.txt">Table of n, a(n) for n = 1..10000</a>
%H A385542 Vaclav Kotesovec, <a href="/A385542/a385542.jpg">Graph - the asymptotic ratio (10000 terms)</a>
%H A385542 <a href="/index/Pow#powerful">Index entries for sequences related to powerful numbers</a>.
%F A385542 a(n) = Sum_{d|n, d < n} A112526(d) * d.
%F A385542 a(n) = A183097(n) - A112526(n) * n.
%F A385542 a(n) = 1 if and only if n is either a squarefree number (A005117) > 1 or a square of a prime (A001248), i.e., if and only if n is in A167207 \ {1}.
%F A385542 Dirichlet g.f.: (zeta(s) - 1)* zeta(2*s-2) * zeta(3*s-3) / zeta(6*s-6).	
%F A385542 Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = zeta(3/2)*(zeta(3/2)-1)/(3*zeta(3)) = 1.168033893310319119603... .
%F A385542 More precise asymptotics: Sum_{k=1..n} a(k) ~ (zeta(3/2) - 1)*zeta(3/2)*n^(3/2) / (3*zeta(3)) + 3*zeta(2/3)*(zeta(4/3) - 1)*n^(4/3) / (2*Pi^2) - n/2. - _Vaclav Kotesovec_, Jul 03 2025
%t A385542 f[p_, e_] := (p^(e+1)-1)/(p-1) - p; a[1] = 0; a[n_] := Times @@ f @@@ (fct = FactorInteger[n]) - If[AllTrue[fct[[;;, 2]], # > 1 &], n, 0]; Array[a, 100]
%o A385542 (PARI) a(n) = {my(f = factor(n), s); s = prod(i=1, #f~, (f[i,1]^(f[i,2]+1)-1)/(f[i,1]-1) - f[i,1]); if(n==1 || vecmin(f[,2]) > 1, s -= n); s};
%Y A385542 Cf. A001248, A001694, A005117, A112526, A167207, A183097.
%Y A385542 Cf. A002117, A078434.
%K A385542 nonn,easy
%O A385542 1,8
%A A385542 _Amiram Eldar_, Jul 03 2025