Primitive divisor conjecture with a finite exceptional divisor bound

Let $(a_n)$ be the sequence under consideration, and call a prime divisor of $a_n$ primitive if it does not divide any earlier term $a_k$ with $k<n$.

Primitive divisor conjecture. The term $a_n$ has a primitive prime divisor for all but finitely many $n$. Moreover, there exists a positive integer $M$ such that, whenever $a_n$ does not have a primitive divisor,

$$n\mid M.$$

This is presented as a generalization of Zsigmondy's theorem. The supplied status is unknown, so it remains open in the database.