The arithmetic derivative is a nonlinear derivation on the positive integers which forms a natural analog of the conventional derivative. While exploring solutions to arithmetic differential equations, we stumbled across a curious pattern in the positive integers for which the arithmetic derivative and the Collatz map commute. Here we report on these empirical findings, and prove several analytical results on the form of such numbers. Among these findings is the existence of a family of semiprime numbers which are mapped by the Collatz function to another semiprime having a sum of prime factors which is half of the original semiprime's. We show that this family of semiprimes solves the commutation problem and that the sum of their reciprocals converges.