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%I #7 Dec 27 2023 01:20:24
%S 1,1,1,2,1,1,1,2,1,1,2,1,1,1,4,1,2,1,2,1,1,1,2,1,2,1,1,1,1,1,1,4,1,1,
%T 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 1,1,4,4,1,1,2,1,1,1,1,2,1,2,1,1,1,1,2
%N Product of exponents of prime factorization of the exponentially 2^n-numbers (A138302).
%C The terms of A005361 that are powers of 2 (A000079).
%C The first position of 2^k, for k = 0, 1, ..., is 1, 4, 15, 126, 1134, ..., which is the position of A085629(2^k) in A138302.
%H Amiram Eldar, <a href="/A368473/b368473.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = A005361(A138302(n)).
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (1/d) * Product_{p prime} (1 + Sum_{k>=0} 2^k/p^(2^k)) = 1.504710204899266020302..., where d = A271727 is the asymptotic density of the exponentially 2^n-numbers.
%t f[n_] := Module[{p = Times @@ FactorInteger[n][[;; , 2]]}, If[p == 2^IntegerExponent[p, 2], p, Nothing]]; Array[f, 150]
%o (PARI) lista(kmax) = {my(p); for(k = 1, kmax, p = vecprod(factor(k)[, 2]); if(p >> valuation(p,2) == 1, print1(p, ", ")));}
%Y Cf. A000079, A005361, A085629, A138302, A271727.
%Y Cf. A366538, A367169.
%Y Similar sequences: A322327, A368472, A368474.
%K nonn,easy
%O 1,4
%A _Amiram Eldar_, Dec 26 2023
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