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A369935 The maximal exponent in the prime factorization of the numbers whose all exponents are squares (A197680).

%I A369935 #11 Mar 29 2025 03:26:51
%S A369935 0,1,1,1,1,1,1,1,1,1,1,4,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,1,
%T A369935 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 A369935 1,1,1,1,1,1,4,1,1,1,1,1,1,1,1,1,1,1,1
%N A369935 The maximal exponent in the prime factorization of the numbers whose all exponents are squares (A197680).
%C A369935 Differs from A368474 at n = 1, 834, 4154, 5822, 6417, ... .
%H A369935 Amiram Eldar, <a href="/A369935/b369935.txt">Table of n, a(n) for n = 1..10000</a>
%F A369935 a(n) = A051903(A197680(n)).
%F A369935 a(n) = A369936(n)^2.
%F A369935 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} (k^2 * (d(k) - d(k-1))) / A357016 = 1.16184898017948977008..., where d(k) = Product_{p prime} (1 - 1/p^2 + Sum_{i=2..k} (1/p^(i^2)-1/p^(i^2+1))) for k >= 1, and d(0) = 0.
%t A369935 squareQ[n_] := IntegerQ[Sqrt[n]]; f[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[AllTrue[e, squareQ], Max @@ e, Nothing]]; f[1] = 0; Array[f, 150]
%o A369935 (PARI) lista(kmax) = {my(e, q); print1(0, ", "); for(k = 2, kmax, e = factor(k)[, 2]; q = 1; for(i = 1, #e, if(!issquare(e[i]), q = 0; break)); if(q, print1(vecmax(e), ", ")));}
%Y A369935 Cf. A000290, A051903, A197680, A357016, A368474, A369936.
%K A369935 nonn,easy
%O A369935 1,12
%A A369935 _Amiram Eldar_, Feb 06 2024