A369933 - OEIS

%I #8 Feb 06 2024 08:14:47

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

%C Differs from A368473 at n = 1, 32, 89, 126, 159, ... .

%H Amiram Eldar, <a href="/A369933/b369933.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A051903(A138302(n)).

%F a(n) = 2^A369934(n), for n >= 2.

%F 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 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 (PARI) ispow2(n) = n >> valuation(n, 2) == 1;

%o lista(kmax) = {my(e); print1(0, ", "); for(k = 2, kmax, e = factor(k)[, 2]; if(ispow2(vecprod(e)), print1(vecmax(e), ", "))); }

%Y Cf. A013661, A051903, A271727, A138302, A368473, A369934.

%K nonn,easy

%O 1,4

%A _Amiram Eldar_, Feb 06 2024