Let

$$F(n) = \max_{\substack{m<n\\ m\textrm{ composite}}} m+p(m),$$

where $p(m)$ is the least prime divisor of $m$. Is it true that $F(n)>n$ for all sufficiently large $n$? Does $F(n)-n\to \infty$ as $n\to\infty$?