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A374325 The maximal exponent in the prime factorization of the numbers whose maximal exponent in their prime factorization is a square.

%I A374325 #7 Jul 06 2024 05:28:26
%S A374325 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 A374325 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 A374325 1,1,1,1,1,1,4,1,1,1,1,1,1,1,1,1,1,1,1
%N A374325 The maximal exponent in the prime factorization of the numbers whose maximal exponent in their prime factorization is a square.
%C A374325 First differs from A369935 at n = 95.
%C A374325 The first occurrence of k^2, for k = 0, 1, ..., is at 1, 2, 12, 335, 42563, ..., which is the position of 2^(k^2) at A369937.
%H A374325 Amiram Eldar, <a href="/A374325/b374325.txt">Table of n, a(n) for n = 1..10000</a>
%F A374325 a(n) = A374326(n)^2.
%F A374325 a(n) = A051903(A369937(n)).
%F A374325 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} k^2 * d(k) / Sum_{k>=1} d(k) = 1.19949058780036416105..., where d(k) = 1/zeta(k^2+1) - 1/zeta(k^2) for k>=2, and d(1) = 1/zeta(2).
%t A374325 f[n_] := Module[{e = If[n == 1, 0, Max[FactorInteger[n][[;; , 2]]]]}, If[IntegerQ@ Sqrt[e], e, Nothing]]; Array[f, 200]
%o A374325 (PARI) lista(kmax) = {my(e); print1(0, ", "); for(k = 2, kmax, e = vecmax(factor(k)[, 2]); if(issquare(e), print1(e, ", ")));}
%Y A374325 Cf. A051903, A369935, A369937.
%Y A374325 Similar sequences: A374324, A374326, A374327, A374328.
%K A374325 nonn,easy
%O A374325 1,12
%A A374325 _Amiram Eldar_, Jul 04 2024