%I #7 Dec 27 2023 01:19:55

%S 1,1,1,1,1,1,1,1,1,1,1,4,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,1,

%T 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,1,1,1,1,1,1,1,1,1,1,1,1,1,1,

%U 1,1,1,1,1,1,4,1,1,1,1,1,1,1,1,1,1,1,1

%N Product of exponents of prime factorization of the numbers whose exponents in their prime power factorization are squares (A197680).

%C All the terms are squares (A000290).

%C The first position of k^2, for k = 1, 2, ..., is 1, 12, 331, 834, 21512290, 26588, ..., which is the position of A085629(k^2) in A197680.

%H Amiram Eldar, <a href="/A368474/b368474.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A005361(A197680(n)).

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (1/d) * Product_{p prime} (1 + Sum_{k>=1} k^2/p^(k^2)) = 1.16776748073813763932..., where d = A357016 is the asymptotic density of A197680.

%t f[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[AllTrue[e, IntegerQ[Sqrt[#]] &], Times @@ e, Nothing]]; Array[f, 150]

%o (PARI) lista(kmax) = {my(e, ok); for(k = 1, kmax, e = factor(k)[, 2]; ok = 1; for(i = 1, #e, if(!issquare(e[i]), ok = 0; break)); if(ok, print1(vecprod(e), ", ")));}

%Y Cf. A000290, A005361, A085629, A197680, A357016.

%Y Similar sequences: A322327, A368472, A368473.

%K nonn,easy

%O 1,12

%A _Amiram Eldar_, Dec 26 2023