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A374586 The maximum exponent in the prime factorization of the numbers that are divisible by the maximum exponent in their prime factorization.

%I A374586 #12 Jul 14 2024 08:51:29
%S A374586 1,1,2,1,1,1,1,1,2,1,1,1,4,1,2,1,2,1,1,1,3,1,3,2,1,1,1,1,1,1,2,1,1,1,
%T A374586 1,1,1,2,1,1,4,2,1,2,1,3,1,1,1,1,2,1,1,1,1,1,2,1,1,1,3,1,1,2,1,1,1,4,
%U A374586 1,1,2,1,1,1,1,2,1,2,1,1,1,1,2,2,1,1,1
%N A374586 The maximum exponent in the prime factorization of the numbers that are divisible by the maximum exponent in their prime factorization.
%H A374586 Amiram Eldar, <a href="/A374586/b374586.txt">Table of n, a(n) for n = 1..10000</a>
%F A374586 a(n) = A051903(A336064(n)).
%F A374586 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} k * d(k) / Sum_{k>=1} d(k) = 1.46254458894503050564..., where d(k) = (1/zeta(k+1)) * Product_{p prime, p|k} ((p^(k-e(p,k)+1) - 1)/(p^(k+1) - 1)) - (1/zeta(k)) * Product_{p prime, p|k} ((p^(k-e(p,k)) - 1)/(p^k - 1)) for k >= 2, and d(1) = 1/zeta(2), and e(p,k) is the exponent of the largest of power of p that divides k.
%t A374586 f[n_] := Module[{e = If[n == 1, 0, Max[FactorInteger[n][[;; , 2]]]]}, If[e > 0 && Divisible[n, e], e, Nothing]]; Array[f, 150]
%o A374586 (PARI) lista(kmax) = {my(e); for(k = 2, kmax, e = vecmax(factor(k)[, 2]); if(!(k % e), print1(e, ", ")));}
%Y A374586 Cf. A051903, A336064, A336065, A374587.
%K A374586 nonn,easy
%O A374586 1,3
%A A374586 _Amiram Eldar_, Jul 12 2024