login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

Product of exponents of prime factorization of the exponentially 2^n-numbers (A138302).
6

%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