Let $\mathbb{N}$ denote the set of positive integers. For $n\in\mathbb{N}$ let $\mathbf{P}_n$ be the set of all positive integers $k$ such that there are at most $n$ different prime numbers that divide $k$. For $A\subseteq \mathbb{N}$ set $$\mu^{+}(A)= \lim \sup_{m\to\infty}\frac{|A \cap\{1,\ldots,m\}|}{m+1}.$$

What (if any) is the smallest $n\in\mathbb{N}$ such that $\mu^+(\mathbf{P}_n) > 0$?