8. If a positive integer $n$ makes the equation $x^{3}+y^{3}=z^{n}$ have positive integer solutions $(x, y, z)$, then $n$ is called a "good number". Then, the number of good numbers not exceeding 2,019 is $\qquad$ .
8.1346.

It is easy to know that if $n$ is a good number, then $n+3$ is also a good number.
Obviously, $1$ and $2$ are good numbers, so when $n \equiv 1,2(\bmod 3)$, $n$ are all good numbers.

By Fermat's Last Theorem, we know that $x^{3}+y^{3}=z^{3}$ has no positive integer solutions, i.e., 3 is not a good number.

Therefore, $n=3 k\left(k \in \mathbf{Z}_{+}\right)$ are not good numbers. Otherwise, if there exists $k \in \mathbf{Z}_{+}$ such that $3 k$ is a good number, i.e., the equation $x^{3}+y^{3}=z^{3 k}$ has a positive integer solution $\left(x_{0}, y_{0}, z_{0}\right)$, then $x^{3}+$ $y^{3}=z^{3}$ has a positive integer solution $\left(x_{0}, y_{0}, z_{0}^{k}\right)$, which is a contradiction.

Hence, $n$ is a good number if and only if $n \equiv 1,2(\bmod 3)$.
Therefore, the number of good numbers not exceeding 2019 is
$$
2019 \times \frac{2}{3}=1346 \text {. }
$$