%I #7 Dec 09 2023 07:09:09

%S 4,8,9,4,16,9,4,8,25,27,4,32,36,8,4,9,16,49,25,4,27,8,4,9,64,4,72,25,

%T 4,16,81,4,8,9,4,32,49,9,100,8,108,16,4,9,8,121,4,125,9,128,4,27,8,4,

%U 144,49,4,25,8,9,4,32,81,4,8,169,9,4,25,16,36,8,4,27

%N The powerful part of the nonsquarefree numbers.

%C The terms of A057521 that are larger than 1, since A057521(k) = 1 if and only if k is squarefree (A005117).

%H Amiram Eldar, <a href="/A368040/b368040.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A057521(A013929(n)).

%F Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = d/(3*(1-1/zeta(2))^(3/2)) = 4.778771..., and d = A328013.

%t f[p_, e_] := If[e > 1, p^e, 1]; powPart[n_] := Times @@ f @@@ FactorInteger[n]; Select[Array[powPart, 200], # > 1 &]

%o (PARI) lista(kmax) = {my(p, f); for(k = 1, kmax, f = factor(k); p = prod(i=1, #f~, if(f[i, 2] > 1, f[i, 1]^f[i, 2], 1)); if(p > 1, print1(p, ", ")));}

%Y Cf. A005117, A013929, A057521.

%Y Cf. A084936, A174961, A275699, A368038, A368039.

%Y Cf. A013661, A229099, A328013.

%K nonn

%O 1,1

%A _Amiram Eldar_, Dec 09 2023