Pizza Toppings

 Published on Friday, 5th March 2010, 01:00 pm; Solved by 905;
Difficulty rating: 55%

Problem 281

You are given a pizza (perfect circle) that has been cut into m·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 ≥ 2), using each topping on exactly n slices (n ≥ 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) ≤ 1015.