Problem 457

Erdős Problem #457: declared status 'proved'. Formalized: yes. Is there some $\epsilon>0$ such that there are infinitely many $n$ where all primes $p\leq (2+\epsilon)\log n$ divide$$\prod_{1\leq i\leq \log n}(n+i)?$$ Current best: More generally, let $q(n,k)$ denote the least prime which does not divide $\prod_{1\leq i\leq k}(n+i)$. Prize: no. OEIS: A391668. Tags: number theory.