Erdős Problem 700: I

Let $f(n) = \min_{1 < k \le n/2} \gcd(n, \binom{n}{k})$ and let $P(n)$ be the largest prime dividing $n$. (a)* Characterise those composite $n$ such that $f(n) = n/P(n)$.

Erdős–Szekeres [ErSz78] note that $f(n) = n/P(n)$ when $n$ is a product of two primes (erdos_700.variants.prime_mul), with $n = 30$ a further example. The characterisation itself is open; we state it as the (unknown) predicate that is equivalent to being such an n.