CZ-SK-MO 2009 Problem 4

A positive integer $n$ is called good if and only if there exist exactly $4$ positive integers $k_1, k_2, k_3, k_4$ such that $n+k_i|n+k_i^2$ ( $1 \leq k \leq 4$ ). Prove that:

• $58$ is good;
• $2p$ is good if and only if $p$ and $2p+1$ are both primes ( $p>2$ ).