  Given an integer $k\ge2$, let $\omega_k(n)$ denote the number of primes that
divide $n$ with multiplicity exactly $k$. We compute the density $e_{k,m}$ of
those integers $n$ for which $\omega_k(n)=m$ for every integer $m\ge0$. We also
show that the generating function $\sum_{m=0}^\infty e_{k,m}z^m$ is an entire
function that can be written in the form $\prod_{p}
\bigl(1+{(p-1)(z-1)}/{p^{k+1}} \bigr)$; from this representation we show how to
both numerically calculate the $e_{k,m}$ to high precision and provide an
asymptotic upper bound for the $e_{k,m}$. We further show how to generalize
these results to all additive functions of the form $\sum_{j=2}^\infty a_j
\omega_j(n)$; when $a_j=j-1$ this recovers a classical result of R\'enyi on the
distribution of $\Omega(n)-\omega(n)$.
