Problem Archive

You are given a pizza (perfect circle) that has been cut into $m \cdot n$ equal pieces and you want to have exactly one topping on each slice.

Let $f(m, n)$ denote the number of ways you can have toppings on the pizza with $m$ different toppings ($m \ge 2$), using each topping on exactly $n$ slices ($n \ge 1$).
Reflections are considered distinct, rotations are not.

Thus, for instance, $f(2,1) = 1$, $f(2, 2) = f(3, 1) = 2$ and $f(3, 2) = 16$.
$f(3, 2)$ is shown below:

Find the sum of all $f(m, n)$ such that $f(m, n) \le 10^{15}$.