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A369717 The sum of divisors of the smallest powerful number that is a multiple of n.

%I A369717 #8 Jan 30 2024 04:44:17
%S A369717 1,7,13,7,31,91,57,15,13,217,133,91,183,399,403,31,307,91,381,217,741,
%T A369717 931,553,195,31,1281,40,399,871,2821,993,63,1729,2149,1767,91,1407,
%U A369717 2667,2379,465,1723,5187,1893,931,403,3871,2257,403,57,217,3991,1281,2863
%N A369717 The sum of divisors of the smallest powerful number that is a multiple of n.
%H A369717 Amiram Eldar, <a href="/A369717/b369717.txt">Table of n, a(n) for n = 1..10000</a>
%F A369717 a(n) = A000203(A197863(n)).
%F A369717 Multiplicative with a(p) = p^2 + p + 1 and a(p^e) = (p^(e+1)-1)/(p-1) for e >= 2.
%F A369717 a(n) >= A000203(n), with equality if and only if n is powerful (A001694).
%F A369717 Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 + 1/p^(s-2) - 1/p^(2*s-3) - 1/p^(2*s-2) + 1/p^(3*s-3)).
%F A369717 Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = zeta(2) * zeta(3) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^5 + 1/p^6 - 1/p^7) = 1.01304866467771286896... .
%t A369717 f[p_, e_] := If[e == 1, p^2 + p + 1, (p^(e + 1) - 1)/(p - 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A369717 (PARI) a(n) = {my(f = factor(n)); for(i = 1, #f~, if(f[i,2] == 1, f[i,2] = 2)); sigma(f);}
%Y A369717 Cf. A000203, A001694, A197863, A369716, A369718, A369720.
%Y A369717 Cf. A002117, A013661, A183699.
%K A369717 nonn,easy,mult
%O A369717 1,2
%A A369717 _Amiram Eldar_, Jan 30 2024