A374328 - OEIS

%I #7 Jul 06 2024 05:29:52

%S 0,0,1,0,0,0,1,0,0,1,0,0,0,2,0,1,0,1,0,0,0,1,0,1,0,0,0,0,0,0,1,0,0,0,

%T 0,0,0,1,1,0,0,2,1,1,0,1,0,0,0,0,0,1,0,0,1,0,0,0,1,0,0,0,0,0,1,1,0,0,

%U 0,2,2,0,0,1,0,0,0,0,1,0,1,0,0,0,0,1,1

%N The base-2 logarithm of the maximal exponent in the prime factorization of the numbers whose maximal exponent in their prime factorization is a power of 2.

%C First differs from {A369934(n+1), n>=1} at n = 378.

%C The first occurrence of k = 0, 1, ... is at 1, 3, 14, 224, 57307, ..., which is the position of 2^(2^k) at A369938.

%H Amiram Eldar, <a href="/A374328/b374328.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = log_2(A374327(n)).

%F a(n) = log_2(A051903(A369938(n))).

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=0} k * d(k) / Sum_{k>=0} d(k) = 0.35575339438419889972..., where d(k) = 1/zeta(2^k+1) - 1/zeta(2^k) for k>=1, and d(0) = 1/zeta(2).

%t f[n_] := Module[{e = If[n == 1, 0, Max[FactorInteger[n][[;; , 2]]]]}, If[e == 2^IntegerExponent[e, 2], IntegerExponent[e, 2], Nothing]]; Array[f, 150]

%o (PARI) lista(kmax) = {my(e); for(k = 2, kmax, e = vecmax(factor(k)[, 2]); if(e >> valuation(e, 2) == 1, print1(valuation(e, 2), ", ")));}

%Y Cf. A051903, A369934, A369938.

%Y Similar sequences: A374324, A374325, A374326, A374327.

%K nonn,easy

%O 1,14

%A _Amiram Eldar_, Jul 04 2024