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Let $q(n,k)$ denote the least prime which does not divide $\prod_{1\leq i\leq k}(n+i)$. Is it true that, if $k$ is fixed and $n$ is sufficiently large, we have \[q(n,k)<(1+o(1))\log n?\]
A problem of Erdős and Pomerance.

The bound $q(n,k)<(1+o(1))k\log n$ is easy. It may be true this improved bound holds even up to $k=o(\log n)$.

See also [457].