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

1,16

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

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)).

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A016406 A386259 A370078 * A372332 A129182 A116857

Adjacent sequences: A386255 A386256 A386257 * A386259 A386260 A386261

Keyword

nonn,easy

Author

Amiram Eldar, Jul 17 2025

Status

approved