The starting number has to be a prime, and the result is that you would cycle to that number and no further.
Alternative result would mean literal mathematical chaos. The number would be an origin point of an attractor. Approaches to find such a counterexample would require something akin to guessing a fractal.
Trying to evaluate the Lyapunov exponent of this system could be fun. Applying Poincare-Bendixson and Bendixson-Dulac to a dual system that is so defined as to reject the final limit cycle and is differentiable, perhaps... Too big math for me. It smells of Hilbert's sixteenth problem.
