Erdős problem 700

Let $f(n) = \min_{1 < k \le n/2} \gcd(n, \binom{n}{k})$.

(b) Are there infinitely many composite $n$ such that $f(n) > n^{1/2}$?

Erdős–Szekeres [ErSz78] could not prove this. (Since $f(n) \ge p(n)$, the least prime factor of $n$, there are infinitely many $n$ — those of the form $p^2$ — with $f(n) \ge n^{1/2}$; the question asks for the strict inequality.) Here $f(n) > n^{1/2}$ is written as (f n) ^ 2 > n.