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A374327 The maximal exponent in the prime factorization of the numbers whose maximal exponent in their prime factorization is a power of 2.

%I A374327 #8 Jul 06 2024 05:29:27
%S A374327 1,1,2,1,1,1,2,1,1,2,1,1,1,4,1,2,1,2,1,1,1,2,1,2,1,1,1,1,1,1,2,1,1,1,
%T A374327 1,1,1,2,2,1,1,4,2,2,1,2,1,1,1,1,1,2,1,1,2,1,1,1,2,1,1,1,1,1,2,2,1,1,
%U A374327 1,4,4,1,1,2,1,1,1,1,2,1,2,1,1,1,1,2,2
%N A374327 The maximal exponent in the prime factorization of the numbers whose maximal exponent in their prime factorization is a power of 2.
%C A374327 First differs from {A369933(n+1), n>=1} at n = 378.
%C A374327 The first occurrence of 2^k, for k = 0, 1, ..., is at 1, 3, 14, 224, 57307, ..., which is the position of 2^(2^k) at A369938.
%H A374327 Amiram Eldar, <a href="/A374327/b374327.txt">Table of n, a(n) for n = 1..10000</a>
%F A374327 a(n) = 2^A374328(n).
%F A374327 a(n) = A051903(A369938(n)).
%F A374327 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=0} 2^k * d(k) / Sum_{k>=0} d(k) = 1.41151462942556759486..., where d(k) = 1/zeta(2^k+1) - 1/zeta(2^k) for k>=1, and d(0) = 1/zeta(2).
%t A374327 f[n_] := Module[{e = If[n == 1, 0, Max[FactorInteger[n][[;; , 2]]]]}, If[e == 2^IntegerExponent[e, 2], e, Nothing]]; Array[f, 150]
%o A374327 (PARI) lista(kmax) = {my(e); for(k = 2, kmax, e = vecmax(factor(k)[, 2]); if(e >> valuation(e, 2) == 1, print1(e, ", ")));}
%Y A374327 Cf. A051903, A369933, A369938.
%Y A374327 Similar sequences: A374324, A374325, A374326, A374328.
%K A374327 nonn,easy
%O A374327 1,3
%A A374327 _Amiram Eldar_, Jul 04 2024