Erdős Problem 430

### What is the conjecture

https://www.erdosproblems.com/430

Fix some integer $n$ and define a decreasing sequence in $[1,n)$ by $a_1=n-1$ and, for $k\geq 2$, letting $a_k$ be the greatest integer in $[1,a_{k-1})$ such that all of the prime factors of $a_k$ are $>n-a_k$.

Is it true that, for sufficiently large $n$, not all of this sequence can be prime?

Status: open

### Choose either option
- [ ] I plan on working on this conjecture
- [x] This issue is up for grabs: I would like to see this conjecture added by somebody else