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A365550 The number of square coreful divisors of n.

%I A365550 #8 Sep 10 2023 02:04:16
%S A365550 1,0,0,1,0,0,0,1,1,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,2,0,0,
%T A365550 0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,
%U A365550 0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0
%N A365550 The number of square coreful divisors of n.
%C A365550 First differs from A188585 at n = 64.
%C A365550 A coreful divisor d of a number n is a divisor with the same set of distinct prime factors as n.
%H A365550 Amiram Eldar, <a href="/A365550/b365550.txt">Table of n, a(n) for n = 1..10000</a>
%F A365550 Multiplicative with a(p^e) = floor(e/2).
%F A365550 a(n) > 0 if and only if n is a powerful number (A001694).
%F A365550 Dirichlet g.f.: zeta(s) * zeta(2*s) * Product_{p prime} (1 - 1/p^s + 1/p^(3*s)).
%e A365550 a(16) = 2 since the coreful divisors of 16 are {2, 4, 8, 16}, and 2 of them, 4 and 16, are squares.
%t A365550 f[p_, e_] := Floor[e/2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A365550 (PARI) a(n) = vecprod(apply(x -> x\2, factor(n)[, 2]));
%Y A365550 Cf. A001694, A005361 (number of coreful divisors), A046951 (number of square divisors), A325837.
%K A365550 nonn,easy,mult
%O A365550 1,16
%A A365550 _Amiram Eldar_, Sep 08 2023