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 $\mu^+(\mathbf{P}\_n) > 0$?



Number of prime factors and density
Let $E$ be the set of positive integers $k$ such that $k$ has $o(\log\log(k))$ distinct prime factors. Then the [Hardy-Ramanujan Theorem](https://en.wikipedia.org/wiki/Hardy%E2%80%93Ramanujan_theorem) implies that $\mu^+(E)=0$. For any $n\geq 1$, $\mathbf{P}\_n\backslash E$ is finite, and so $\mu^+(\mathbf{P}\_n)=0$. 


