# Infinite subset of $\mathbb{N}$ almost avoiding all “zebra crossings”

Given a positive integer $n\in\mathbb{N}$ we define the zebra crossing associated with $n$ to be the set $$Z_n = \{[2kn, 2kn+(n-1)] \cap \mathbb{N}: k\in\mathbb{N}\}.$$

Is there an infinite set $A\subseteq\mathbb{N}$ such that $A\cap Z_n$ is finite for all positive integers $n$?

Take the $k$th element of $A$ to be $a_k:=2k!-1$, for all $k\ge 1$. If $k\ge n$, then $a_k\equiv -1\!\!\pmod{2n}$, whence $a_k\notin Z_n$. Thus, for any fixed $n$, there are only finitely many elements of $A$ contained in $Z_n$.