Erdős Problem #962

declared status 'open'. Formalized: yes. Let $k(n)$ be the maximal $k$ such that there exists $m\leq n$ such that each of the integers$$m+1,\ldots,m+k$$are divisible by at least one prime $>k$. Estimate $k(n)$. Current best: Erd\H{o}s [Er65] wrote it is 'not hard to prove' that$$k(n)\gg_\epsilon \exp((\log n)^{1/2-\epsilon})$$and it 'seems likely' that $k(n)=o(n^\epsilon)$, but had no non-trivial upper bound for $k(n)$. Tang has proved a lower bound of$$k(n)\geq \exp\left(\left(\frac{1}{\sqrt{2}}-o(1)\right)\sqrt{\log n\log\log n}\right).$$ References [Er65] Erd\H{o}s, P., Extremal problems in number theory. Prize: no. OEIS: A327909. Tags: number theory.

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