OPEN - $78

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 $f(n)$.

A problem of Erdős and Pomerance [ErPo80], who proved
\[n\left(\frac{\log n}{\log\log n}\right)^{1/2}\ll f(n)\leq (2+o(1))n(\log n)^{1/2}.\]

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].