Erdős Problem #493

Does there exist a \(k\) such that every sufficiently large integer can be written in the form \[\prod_{i=1}^k a_i - \sum_{i=1}^k a_i\] for some integers \(a_i\geq 2\)? Erdős attributes this question to Schinzel. Eli Seamans has observed that the answer is yes (with \(k=2\)) for a very simple reason: \(n = 2(n+2)-(2+(n+2))\). There may well have been some additional constraint in the problem as Schinzel posed it, but [Er61] does not record what this is.