Note that trivially $F(n) \leq n + \sqrt{n}$.

Let Unparsed notation: F(n) := \max\{m + p(m) \mid \textrm{m < nUnparsed notation: composite}\}\} where $p(m)$ is the least prime divisor of $m$. Is it true that $F(n)>n$ for all sufficiently large $n$?