OPEN
Let $\epsilon>0$ and $n$ be large depending on $\epsilon$. Is it true that for all $n^\epsilon<k\leq n^{1-\epsilon}$ the number of distinct prime divisors of $\binom{n}{k}$ is
\[(1+o(1))k\sum_{k<p<n}\frac{1}{p}?\]
Or perhaps even when $k \geq (\log n)^c$?
It is trivial that the number of prime factors is
\[>\frac{\log \binom{n}{k}}{\log n},\]
and this inequality becomes (asymptotic) equality if $k>n^{1-o(1)}$.
