A386258 Exponent of the highest power of 2 dividing the product of exponents of the prime factorization of n.

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

Offset: 1

Author

Amiram Eldar, Jul 17 2025

Keywords

• nonn
• easy

Comments

First differs from A386259 at n = 36.

First differs from A370078 at n = 64.

The first occurrence of k = 0, 1, 2, ... is at n = A085629(2^k) = 1, 4, 16, 144, 1296, 20736, 518400, ... .

The asymptotic density of the occurrences of 1 in this sequence is the asymptotic density of numbers whose prime factorization has only odd exponents except for one exponent that is of the form 4*k+2 (k >= 0) which equals A065463 * Sum_{p prime} p^2/(p^4+p^3+p-1) = 0.22670657681840536721... .

Links

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

Crossrefs

Cf. A005361, A007814, A065463, A085629, A268335, A295664, A370078, A386259.

Programs

Mathematica

a[n_] := IntegerExponent[Times @@ FactorInteger[n][[;; , 2]], 2]; Array[a, 100]

PARI

a(n) = vecsum(apply(x -> valuation(x, 2), factor(n)[, 2]));

Formula

a(n) = A007814(A005361(n)).

Additive with a(p^e) = A007814(e).

a(n) = 0 if and only if n is an exponentially odd number (A268335).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} f(1/p) = 0.37572872586497617473..., where f(x) = Sum_{k>=1} x^(2^k)/(1-x^(2^k)).