Compute $f(n)=\min_{1<k\le n/2}\gcd(n,\binom{n}{k})$: composite $n$ with $f(n)>\sqrt{n}$ (Erdős #700)

For a positive integer $n$ define $f(n)=\min_{1<k\le n/2}\gcd\!\left(n,\binom{n}{k}\right)$. Erdős and Szekeres posed three questions about composite $n$: (a) characterise the composite $n$ with $f(n)=n/P(n)$, where $P(n)$ is the largest prime factor of $n$; (b) are there infinitely many composite $n$ with $f(n)>n^{1/2}$? (c) is it true that $f(n)\ll_A n/(\log n)^A$ for every $A>0$? Each $\gcd(n,\binom{n}{k})$ is computable exactly (via Kummer's theorem: $v_p\binom{n}{k}$ = number of carries adding $k$ and $n-k$ in base $p$), so $f(n)$ is a finite minimization. GOAL (computational): compute $f(n)$ for all composite $n\le N$, extract the set with $f(n)>\sqrt n$, test the density hypotheses in (a)-(c), and extend the associated data.

Acceptance. PARTIAL / PRIMARY (finite, machine-checkable): a table of $f(n)$ for all composite $n\le N$ (state $N$), computed via Kummer/Legendre valuations, delivered with (i) a runnable script, (ii) the list of composite $n\le N$ with $f(n)>\sqrt n$ (extending OEIS A091963 where applicable), and (iii) an empirical characterization of the $n$ with $f(n)=n/P(n)$ plus a fit testing $f(n)\lesssim n/(\log n)^A$. Cross-check known A091963 terms. FULLY RESOLVES: a proof answering (b) or (c). State $N$ and the valuation method used.