The $n$-th Mersenne number $M_n$ is defined as $$M_n=2^n-1$$ A great deal of research focuses on Mersenne primes. What is known in the opposite direction about Mersenne numbers with only small factors (i.e. smooth numbers)? In particular, if we let $P_n$ denote the largest prime factor of $M_n$, are any results known of the form $$\liminf_{n\rightarrow \infty}\frac{P_n}{f(n)}= 1$$ for some function $f$?

I've only come across two (fairly distant) bounds so far. If we consider even-valued $n$, then $M_n=M_{n/2}(M_{n/2}+2)$, so: $$\liminf_{n\rightarrow \infty}\frac{P_n}{2^{n/2}}\leq 1$$ In the other direction, [1] shows that $P_n\geq 2n+1$ for $n>12$, so $$\liminf_{n\rightarrow \infty}\frac{P_n}{2n}\geq 1$$

[1] A. Schinzel, On primitive prime factors of $a^n-b^n$, Proc. Cambridge Philos. Soc. 58 (1962), 555-562.