Problem 710

Erdős Problem #710: declared status 'open'. Formalized: no. Let $f(n)$ be minimal such that in $(n,n+f(n))$ there exist distinct integers $a_1,\ldots,a_n$ such that $k\mid a_k$ for all $1\leq k\leq n$. Obtain an asymptotic formula for $f(n)$. Current best: A problem of Erd\H{o}s and Pomerance [ErPo80], who proved$$(2/\sqrt{e}+o(1))n\left(\frac{\log n}{\log\log n}\right)^{1/2}\leq f(n)\leq (1.7398\cdots+o(1))n(\log n)^{1/2}.$$In [Er92c] Erd\H{o}s offered 2000 rupees for an asymptotic formula; for uniform comparison across prizes I have converted this using the 1992 exchange rates. Prize: ₹2000. OEIS: A390246. Tags: number theory.