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 2019 is $\qquad$ .
$$
\begin{array}{l}
1^{3}+1^{3}=2 \Rightarrow 2^{3}+2^{3}=2^{4} \Rightarrow 4^{3}+4^{3}=2^{7}, 8^{3}+8^{3}=2^{10}, \cdots, \\
2^{3}+2^{3}=4^{2} \Rightarrow 8^{3}+8^{3}=4^{5} \Rightarrow 32^{3}+32^{3}=4^{8}, \cdots
\end{array}
$$

By Fermat's Last Theorem, $x^{3}+y^{3}=z^{3}$ has no positive integer solutions, so except for multiples of 3, $x^{3}+y^{3}=z^{n}$ has positive integer solutions. Therefore, the number of "good numbers" is $\frac{2}{3} \times 2019=1346$.