Problem 700

Erdős Problem #700: declared status 'open'. Formalized: no. Let$$f(n)=\min_{1<k\leq n/2}\textrm{gcd}\left(n,\binom{n}{k}\right).$${UL} {LI}Characterise those composite $n$ such that $f(n)=n/P(n)$, where $P(n)$ is the largest prime dividing $n$.{/LI} {LI}Are there infinitely many composite $n$ such that $f(n)>n^{1/2}$?{/LI} {LI} Is it true that, for every composite $n$,$$f(n) \ll_A \frac{n}{(\log n)^A}$$for every $A>0$?{/LI} {/UL} Current best: This implies$$f(n) \leq (1+o(1))\frac{n}{\log n}.$$It is known that $f(n)=n/P(n)$ when $n$ is the product of two primes. Prize: no. OEIS: A091963. Tags: binomial coefficients, number theory.