Problem 389

Erdős Problem #389: declared status 'open'. Formalized: yes. Is it true that for every $n\geq 1$ there is a $k$ such that$$n(n+1)\cdots(n+k-1)\mid (n+k)\cdots (n+2k-1)?$$ Current best: For example when $n=2$ we have $k=5$:$$2\times 3 \times 4 \times 5\times 6 \mid 7 \times 8 \times 9\times 10\times 11.$$and when $n=3$ we have $k=4$:$$3\times 4\times 5\times 6 \mid 7\times 8\times 9\times 10.$$Bhavik Mehta has computed the minimal such $k$ for $1\leq n\leq 18$ (now available as A375071 on the OEIS). Prize: no. OEIS: A375071. Tags: number theory.