Many thanks to Cameron Croucher for contributing this week's puzzle.

A function F is defined from the positive integers to the positive integers.

It satisfies two properties:

(i) F(ab) = F(a)F(b) whenever a and b have no common factor greater than 1.

(ii) F(p + q) = F(p) + F(q) for any two primes p and q (which might be the same).

 

Prove that F(2) = 2, F(3) = 3, and F(2015) = 2015. Can you find F(1999)?