A374328 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.

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, 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, 0, 2, 2, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1

Offset

1,14

Comments

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

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

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

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

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

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).

Mathematica

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]

Prog

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

Crossrefs

Cf. A051903, A369934, A369938.

Similar sequences: A374324, A374325, A374326, A374327.

Sequence in context: A129182 A116857 A369934 * A372505 A322338 A158971

Adjacent sequences: A374325 A374326 A374327 * A374329 A374330 A374331

Keyword

nonn,easy

Author

Amiram Eldar, Jul 04 2024

Status

approved