Problem 693

Erdős Problem #693: declared status 'open'. Formalized: no. Let $k\geq 2$ and $n$ be sufficiently large depending on $k$. Let $A=\{a_1<a_2<\cdots \}$ be the set of those integers in $[n,n^k]$ which have a divisor in $(n,2n)$. Estimate$$\max_{i} a_{i+1}-a_i.$$Is this $\leq (\log n)^{O(1)}$? Prize: no. OEIS: A391118. Tags: divisors, number theory.