The intermediate prime divisors of integers

Abstract:

Let ${p_1} < {p_2} < \cdots < {p_\omega }$ be the distinct prime divisors of the integer $n$. If $\omega = \omega (n) \to + \infty$ with $n$, then ${p_j}$ is called an intermediate prime divisor of $n$ if both $j$ and $\omega - j$ tend to infinity with $n$. We show that $\log \log {p_j}$, as $j$ goes through the indices for which ${p_j}$ is intermediate, forms a limiting Poisson process in the sense of natural density.