  Let $\{p_j(n)\}_{j=1}^{\omega(n)}$ denote the increasing sequence of distinct
prime factors of an integer $n$. For $z\geqslant 0$, let $G(n;z)$ denote the
number of those indexes $j$ such that $p_{j+1}(n)>p_j(n)^{\exp z}$. We show
uniform convergence, with almost optimal effective estimate of the speed, of
the distribution of $G(n;z)$ on $\{n:1\leqslant n\leqslant N\}$ to a Gaussian
limit law with mean ${\rm e}^{-z}\log_2n$ and variance $\{{\rm e}^{-z}-2z{\rm
e}^{-2z}\}\log_2n$, and we establish an asymptotic formula with remainder for
all centered moments.
