The number of positive integers that are products of a given number of primes follows an asymptotic similar to the prime number theorem. These numbers are called $k$-almost primes.
More precisely, the counting function $\pi_k(x)$ of numbers that are the product of $k$ primes is asymptotically:
$$\frac{1}{(k-1)!} \frac{x}{\log x } (\log \log x)^{k-1} $$
In particular, the density behaves like $\frac{(\log \log x)^{k-1}}{\log x }$ and thus is $0$. For a lot more details see Asymptotic density of k-almost primes
