Problem 800

Hybrid Integers

An integer of the form $p^q q^p$ with prime numbers $p \neq q$ is called a hybrid-integer.
For example, $800 = 2^5 5^2$ is a hybrid-integer.

We define $C(n)$ to be the number of hybrid-integers less than or equal to $n$.

You are given $C(800) = 2$ and $C(800^{800}) = 10790$.

Find $C(800800^{800800})$.

若一个整数可以表示为$p^q q^p$的形式，且$p \neq q$均为质数，则称其为混合整数。
例如，$800 = 2^5 5^2$是一个混合整数。

定义$C(n)$为小于等于$n$的所有混合整数的数目。

已知$C(800) = 2$，$C(800^{800}) = 10790$。

求$C(800800^{800800})$。