How can we decide whether for a given positive integer $k$ , $\varphi(n)+n$ can divide $n^2+k$ , where $\varphi(n)$ denotes the totient function ?
Some cases are easy : $n=1$ is a solution for odd $k$. $n=2$ is the smallest solution for $k\equiv 2\mod 6$
For $k=214$, the smallest solution is $4\ 359\ 549$. For $k=382$, I have not found a solution yet. According to my calculations, there is no solution $n\le 2\cdot 10^9$
Is there any systematic way to find the solutions (or at least one) or to prove that there is none ?