Find a sequence of whole numbers $n _ { 1 } , n _ { 2 } , \ldots$ such that $n _ { i - 1 } | n _ { i }$ for all $i \geq 2$

Question

Problem : Find a sequence of whole numbers $n _ { 1 } , n _ { 2 } , \ldots$ such that $n _ { i - 1 } | n _ { i }$ for all $i \geq 2$ and for every $k \in \mathbb{N}$ there exists $i$ such that $k | n _ { i }$.

The subject of the exercice is "the splitting field of $\mathbb{F}_p$", with $p$ a prime. But this first question doesn't seem to relate (yet) to rings and fields theory.

This question makes me think of the proof of the the fundamental theorem of finite abelian groups with invariant factors.

Answer 1

What about $n_i = i!$?

(Is this enough characters?)