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A369933 The maximal exponent in the prime factorization of the exponentially 2^n numbers (A138302).

%I A369933 #12 Mar 29 2025 03:26:35
%S A369933 0,1,1,2,1,1,1,2,1,1,2,1,1,1,4,1,2,1,2,1,1,1,2,1,2,1,1,1,1,1,1,2,1,1,
%T A369933 1,1,1,1,2,2,1,1,4,2,2,1,2,1,1,1,1,1,2,1,1,2,1,1,1,2,1,1,1,1,1,2,2,1,
%U A369933 1,1,4,4,1,1,2,1,1,1,1,2,1,2,1,1,1,1,2
%N A369933 The maximal exponent in the prime factorization of the exponentially 2^n numbers (A138302).
%C A369933 Differs from A368473 at n = 1, 32, 89, 126, 159, ... .
%H A369933 Amiram Eldar, <a href="/A369933/b369933.txt">Table of n, a(n) for n = 1..10000</a>
%F A369933 a(n) = A051903(A138302(n)).
%F A369933 a(n) = 2^A369934(n), for n >= 2.
%F A369933 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (1/zeta(2) + Sum_{k>=1} (2^k * (d(k) - d(k-1)))) / A271727 = 1.40540547368932408503..., where d(k) = Product_{p prime} (1 - 1/p^3 + Sum_{i=2..k} (1/p^(2^i)-1/p^(2^i+1))) for k >= 1, and d(0) = 1/zeta(2).
%t A369933 pow2Q[n_] := n == 2^IntegerExponent[n, 2]; f[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[AllTrue[e, pow2Q], Max @@ e, Nothing]]; f[1] = 0; Array[f, 150]
%o A369933 (PARI) ispow2(n) = n >> valuation(n, 2) == 1;
%o A369933 lista(kmax) = {my(e); print1(0, ", "); for(k = 2, kmax, e = factor(k)[, 2]; if(ispow2(vecprod(e)), print1(vecmax(e), ", "))); }
%Y A369933 Cf. A013661, A051903, A271727, A138302, A368473, A369934.
%K A369933 nonn,easy
%O A369933 1,4
%A A369933 _Amiram Eldar_, Feb 06 2024