Collatz Stopping Time

The Collatz stopping time of a positive integer is the number of iterations needed for its trajectory under the map in the Collatz conjecture to first fall below . The count depends on whether the ordinary map or the accelerated Terras map is used. For the Terras map, it is

with if the set is empty (Lagarias 1985, p. 5). Thus, , , and , whereas counting ordinary steps gives .

The total Collatz stopping time counts steps until 1 is first reached. Using the ordinary map and counting zero steps when , it is

with if 1 is never reached. Its values for , 2, ... begin 0, 1, 7, 2, 5, 8, 16, 3, 19, 6, ... (OEIS A006577). In contrast, the Terras map has total stopping time 5 for .

The Collatz conjecture is equivalent to finite total stopping time for every positive integer. It is also equivalent to finite first-drop time for every integer , by the principle of strong induction (Lagarias 1985, p. 5).