BALTIC-WAY 2020 Problem 18

Let $n \geqslant 1$ be a positive integer. We say that an integer $k$ is a fan of $n$ if $0 \leqslant k \leqslant n-1$ and there exist integers $x, y, z \in \mathbb{Z}$ such that

$$\begin{aligned} x^{2}+y^{2}+z^{2} & \equiv 0 (\bmod n) ; \\ x y z & \equiv k (\bmod n) . \end{aligned}$$

Let $f(n)$ be the number of fans of $n$. Determine $f(2020)$.