For any positive integer $n$, and $i = 1, 2$, let $f_i(n)$ denote the number of divisors of $n$ of the form $3k + i$ (including 1 and $n$). Define, for any positive integer $n$,
$$f(n) = f_1(n) - f_2(n)$$
Find the values of $f\left(5^{2022}\right)$ and $f\left(21^{2022}\right)$.