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A370329 a(n) is the number of coreful divisors of the n-th powerful number that are also powerful numbers.

%I A370329 #10 Feb 16 2024 02:11:26
%S A370329 1,1,2,1,3,1,2,4,1,1,5,2,3,1,2,1,2,6,3,1,1,2,4,1,4,7,4,1,3,2,1,2,3,6,
%T A370329 1,1,2,8,1,5,3,6,2,1,5,3,4,1,8,1,1,2,4,4,9,1,2,6,1,1,9,2,2,2,1,2,1,1,
%U A370329 4,5,1,10,1,2,1,3,8,6,3,10,1,6,2,1,7,2,3
%N A370329 a(n) is the number of coreful divisors of the n-th powerful number that are also powerful numbers.
%C A370329 A coreful divisor d of a number n is a divisor with the same set of distinct prime factors as n (see A307958).
%C A370329 The positive terms of A361430.
%H A370329 Amiram Eldar, <a href="/A370329/b370329.txt">Table of n, a(n) for n = 1..10000</a>
%F A370329 a(n) = A361430(A001694(n)).
%F A370329 a(n) = 1 if and only if n is the square of a squarefree number (A062503).
%F A370329 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(3/2) * zeta(3) * Product_{p prime} (1 + 2/p^2 + 2/p^(5/2) - 1/p^3 - 2/p^(7/2) - 2/p^4) = 6.91748056612108993003... . (The infinite product of primes is the value of f(1/2) in A361430).
%t A370329 f[p_, e_] := e - 1; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; With[{max = 10^4}, s /@ Union@ Flatten@ Table[i^2*j^3, {j, 1, max^(1/3)}, {i, 1, Sqrt[max/j^3]}]]
%o A370329 (PARI) lista(kmax) = {my(e); for(k = 1, kmax, e = factor(k)[,2]; if(k == 1 || vecmin(e) > 1, print1(prod(i = 1, #e, e[i]-1), ", ")));}
%Y A370329 Cf. A001694, A002117, A062503, A078434, A307958, A360908 (analogous with squares), A361430, A370328.
%K A370329 nonn,easy
%O A370329 1,3
%A A370329 _Amiram Eldar_, Feb 15 2024