Question 6
Let $S$ be a set of positive integers, possibly infinite, such that no positive integer greater than 1 divides all elements of $S$. Determine all non-periodic infinite sequences $a_1, a_2, a_3, \ldots$ of positive integers such that, for all positive integers $n$,
• $ a_n \leq |a_{n+\ell} - \ell| $ for all $ \ell $ in $ S $, and
• $ a_{n} = |a_{n+\ell} - \ell| $ for at least one $ \ell $ in S.
We say that an infinite sequence $a_{1}, a_{2}, a_{3}, \ldots$ is periodic if there exists a positive integer $t$ such that $a_{n} = a_{n+t}$ for all positive integers $n$.
• $ a_n \leq |a_{n+\ell} - \ell| $ for all $ \ell $ in $ S $, and
• $ a_{n} = |a_{n+\ell} - \ell| $ for at least one $ \ell $ in S.
We say that an infinite sequence $a_{1}, a_{2}, a_{3}, \ldots$ is periodic if there exists a positive integer $t$ such that $a_{n} = a_{n+t}$ for all positive integers $n$.