Finding the Best Constant in Prime Counting Function Relation

Question

How close can we approximate the best constant $c$ such that

$n^{\pi(2n)- \pi(n)} \le c^n$ for all positive integers $n$.

I know that $c = 4$ works from $n^{\pi(2n)-\pi(n)} < \prod_{n < p \le 2n }p^n < \dbinom{2n}n < (1+1)^{2n} = 4^n$.

Answer 1

As Gerry wrote, the correct value will be around $e$ by the Prime Number Theorem. For small numbers I can't get any closer than at $n=57$ which still has $c<e$. Probably a value just slightly higher than $e$ would work. It's not possible for $c<e$.