OPEN

Can every integer $N\geq 2$ be written as
\[N=\frac{\prod_{1\leq i\leq k}(m+i)}{\prod_{1\leq i\leq k}(n+i)}\]
for some $k\geq 2$ and $m\geq n+k$?

If $n$ and $k$ are fixed then can one say anything about the set of integers so represented?

See also [677].