1. Among the natural numbers from $1 \sim 10000$, the integers that are neither perfect squares nor perfect cubes are $\qquad$ in number.
In the natural numbers from $1 \sim 10000$, there are 100 perfect squares.
$$
\begin{array}{l}
\text { Because } 22^{3}=10648>10000, \\
21^{3}=9261<10000,
\end{array}
$$

Therefore, there are 21 perfect cubes.
Next, consider the number of natural numbers from $1 \sim 10000$ that are both perfect squares and perfect cubes (i.e., perfect sixth powers). Since $4^{6}=4096,5^{6}=15625$, there are 4 perfect sixth powers.

By the principle of inclusion-exclusion, the number of natural numbers from $1 \sim 10000$ that are neither perfect squares nor perfect cubes is
$$
10000-100-21+4=9883 .
$$