CN-TST 2019 Problem 4

Call a sequence of positive integers $\{a_n\}$ good if for any distinct positive integers $m,n$ , one has $$\gcd(m,n) \mid a_m^2 + a_n^2 \text{ and } \gcd(a_m,a_n) \mid m^2 + n^2.$$ Call a positive integer $a$ to be $k$ -good if there exists a good sequence such that $a_k = a$ . Does there exists a $k$ such that there are exactly $2019$ $k$ -good positive integers?