Problem 711

Erdős Problem #711: declared status 'open'. Formalized: no. 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$$\max_m (f(n,m)-f(n,n))\to \infty.$$ Prize: ₹1000. Tags: number theory.