Distribution of the number of prime factors with a given multiplicity

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 Rényi on the distribution of \(\Omega(n)-\omega(n)\).