Distribution of the number of prime factors with a given multiplicity

Given an integer $k\ge 2$, 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 $m\ge 0$. We also show that the generating function ${\sum }_{m=0}^{\mathrm{\infty }}{e}_{k,m}{z}^m$ is an entire function that can be written in the form ${\prod }_p(1+(p-1)(z-1)\mathrm{/}{p}^{k+1})$; 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 ${\sum }_{j=2}^{\mathrm{\infty }}{a}_j{\omega }_j(n)$; when $a_j=j-1$ this recovers a classical result of Rényi on the distribution of $\mathrm{\Omega }(n)-\omega (n)$.