OPEN

Let $a_1<a_2<\cdots$ be a sequence of integers such that
\[\lim_{n\to \infty}\frac{a_n}{a_{n-1}^2}=1\]
and $\sum\frac{1}{a_n}\in \mathbb{Q}$. Then, for all sufficiently large $n\geq 1$,
\[ a_n = a_{n-1}^2-a_{n-1}+1.\]

A sequence defined in such a fashion is known as Sylvester's sequence.