Distribution of the number of prime factors with a given multiplicity

Given an integer k≥2, let ωₖ(n) denote the number of primes that divide n with multiplicity exactly k. We compute the density ek,m of those integers n for which ωₖ(n)=m for every integer m≥0. We also show that the generating function ∑ₘ₌₀^∞ ek,mzᵐ is an entire function that can be written in the form ∏ₚ (1+(p-1)(z-1)/pᵏ⁺¹ ); from this representation we show how to both numerically calculate the ek,m to high precision and provide an asymptotic upper bound for the ek,m. We further show how to generalize these results to all additive functions of the form ∑ⱼ₌₂^∞ aⱼ ωⱼ(n); when aⱼ=j-1 this recovers a classical result of R\'enyi on the distribution of Ω(n)-ω(n).