Find the number of ordered pairs $(a, b)$ of positive integers such that $a$ and $b$ both divide $20^{19}$, but $a b$ does not.
Write $N=20^{19}=2^{38} \cdot 5^{19}$

The number of divisors of $N$ is $39 \cdot 20=780$.

The number of pairs $(a, b)$ with $a b$ dividing $N$ is equal to the number of nonnegative integer solutions to $x+y \leq 38$ times the number of nonnegative integer solutions to $x+y \leq 19$; by a stars-and-bars argument this equals $\left(\begin{array}{c}40 \\ 2\end{array}\right)\left(\begin{array}{c}21 \\ 2\end{array}\right)=163800$.

Therefore the answer is $780^{2}-163800=444600$.