Erdős Problem #700 — Let $f(n)=\min_{1<k\leq n/2}\textrm{gcd}\left(n,\binom{n}{k}\right).$ - Characterise those composite $n$ such that $f(n)=n/P(n)$, where $P(n)$ is the largest prime dividing $n$.

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

• Characterise those composite $n$ such that $f(n)=n/P(n)$, where $P(n)$ is the largest prime dividing $n$.

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

• Is it true that, for every composite $n$, $f(n) \ll_A \frac{n}{(\log n)^A}$ for every $A>0$?