Abstract. The Collatz conjecture the (3x+1)-problem is considered in the paper. A new form of the Collatz function is proposed, which can be studied analytically. The structure of the core of this function is analyzed by affine automata and used to construct an increasing hierarchy of Bohm–Zontacchi numbers. Successful trajectories of Collatz function iterations are coded by natural numbers and any natural number is shown to be a code of some trajectory.

Keywords: (3x+1)-problem, Collatz conjecture, affine automaton.