The congruence $x^n\equiv -a^n\pmod{m}$: Solvability and related OEIS sequences

We study the solvability of the congruence $x^n\equiv -a^n\pmod{m}$, where $n,m\in\mathbb{Z}_+$, $a\in\mathbb{Z}$, and $\gcd{(a,m)}=1$. Our motivation arises from computer experiments concerning a geometric property of the roots of the congruence $x^n+y^n\equiv 0\pmod{p}$, where $n\in\mathbb{Z}_+$ and $p\in\mathbb{P}$. We encounter several OEIS sequences. We also make new observations on some of them.