Upper Bound and Lower Bound of the Sum of the Prime Divisors of a Odd Semiprime

Question

Lets say we have $n$, an odd semiprime. $p$ and $q$ are odd primes, such that $pq=n$. What are the tightest upper and lower bounds of $p+q$ in terms of $n$ known right now? Right now, I have $2\sqrt n\leq p+q<n$ but I don't know if that's the tightest possible.

Answer 1

Summing up the comments; without loss of generality $3\leq p\leq\sqrt{n}\leq q$, and because $n=pq$ we have $$p+q=p+\tfrac np,$$ which is maximal when $p$ is minimal, and minimal when $p$ is maximal. Hence $$2\sqrt{n}\leq p+q\leq 3+\tfrac n3.$$