Problem Statement
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 $f(n)$.
Categories:
Number Theory
Progress
A problem of Erdős 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] Erdős offered 2000 rupees for an asymptotic formula; for uniform comparison across prizes I have converted this using the 1992 exchange rates.See also [711].
Source: erdosproblems.com/710 | Last verified: January 16, 2026