Let $n = \begin{equation*} \prod_{p \in P}^{} p^{v_p(n)} \end{equation*}$ be a positive integer such that for all p ∈ P we have $ν_p(n)$ ≤ 1. Moreover, $p \in P$ a prime p divides n if and only if p − 1 divides n, too. Compute n.
I came across with this example, but actually have no idea how even to start with it.
Anyone can help?
To start, note that $1 \mid n$ which forces $2\mid n$ since $2$ is prime. Then $3 \mid n$ since $3$ is prime. This forces $6 \mid n$ and thus $7 \mid n$ since $7$ is prime. Can you take this further?
EDIT: Here is the earlier problem that I couldn't locate before: Prove that there exist infinitely many natural numbers $n$ such that $p\mid n$ iff $p-1\mid n$
The criterion is almost identical but it doesn't require $n$ to be squarefree, which yields a few additional solutions. That problem asks to prove there are infinitely many, but no progress was made on that front. On the other hand, of the 4 known examples only one is squarefree.