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A374587 The maximum exponent in the prime factorization of the numbers that are not coprime to the maximum exponent in their prime factorization.

%I A374587 #11 Jul 14 2024 08:51:34
%S A374587 2,2,4,2,2,3,3,2,2,2,4,2,2,3,2,6,2,3,2,4,2,2,2,2,2,3,4,2,3,2,2,2,3,2,
%T A374587 4,2,2,2,5,4,2,3,2,4,2,2,3,6,2,2,2,4,2,3,2,2,2,2,4,2,2,2,8,2,3,2,3,4,
%U A374587 2,2,2,2,3,2,4,2,2,3,2,6,4,2,4,2,2,2,2
%N A374587 The maximum exponent in the prime factorization of the numbers that are not coprime to the maximum exponent in their prime factorization.
%H A374587 Amiram Eldar, <a href="/A374587/b374587.txt">Table of n, a(n) for n = 1..10000</a>
%F A374587 a(n) = A051903(A368715(n)).
%F A374587 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=2} k * d(k) / Sum_{k>=2} d(k) = 2.74240523513766773312..., where d(k) = (1 - 1/f(k+1, k))/zeta(k+1) - (1 - 1/f(k, k))/zeta(k), and f(e, m) = Product_{primes p|m} ((1-1/p^e)/(1-1/p)).
%t A374587 f[n_] := Module[{e = If[n == 1, 0, Max[FactorInteger[n][[;; , 2]]]]}, If[!CoprimeQ[n, e], e, Nothing]]; Array[f, 350]
%o A374587 (PARI) lista(kmax) = {my(e); for(k = 2, kmax, e = vecmax(factor(k)[, 2]); if(gcd(k, e) > 1, print1(e, ", ")));}
%Y A374587 Cf. A051903, A368715, A374586.
%K A374587 nonn,easy
%O A374587 1,1
%A A374587 _Amiram Eldar_, Jul 12 2024