A369934 a(n) = log_2(A369933(n)).

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: 2

Author

Amiram Eldar, Feb 06 2024

Keywords

• nonn
• easy

Links

• Amiram Eldar, Table of n, a(n) for n = 2..10001

Crossrefs

Cf. A013661, A271727, A369933.

Programs

Mathematica

pow2Q[n_] := n == 2^IntegerExponent[n, 2]; f[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[AllTrue[e, pow2Q], Log2[Max @@ e], Nothing]]; Array[f, 150, 2]

PARI

ispow2(n) = n >> valuation(n, 2) == 1;
lista(kmax) = {my(e); for(k = 2, kmax, e = factor(k)[, 2]; if(ispow2(vecprod(e)), print1(logint(vecmax(e), 2), ", "))); }

Formula

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=2..m} a(k) = (Sum_{k>=1} (k * (d(k) - d(k-1)))) / A271727 = 0.35202853155774942465..., 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).