OPEN - $39

Let $f(n,m)$ be minimal such that in $(m,m+f(n,m))$ there exist distinct integers $a_1,\ldots,a_n$ such that $k\mid a_k$ for all $1\leq k\leq n$. Prove that
\[\max_m f(n,m) \leq n^{1+o(1)}\]
and that
\[\max_m (f(n,m)-f(n,n))\to \infty.\]

A problem of Erdős and Pomerance [ErPo80], who proved that
\[\max_m f(n,m) \ll n^{3/2}.\]

In [Er92c] Erdős offered 1000 rupees for a proof of either; for uniform comparison across prizes I have converted this using the 1992 exchange rates.

See also [710].