Title: Clusters of Integers with Equal Total Stopping Times in the 3x + 1 Problem,
Abstract: The clustering of integers with equal total stopping times has long been
observed in the 3x + 1 Problem, and a number of elementary results about it
have been used repeatedly in the literature. In this paper we introduce a
simple recursively defined function C(n), and we use it to give a necessary and
sufficient condition for pairs of consecutive even and odd integers to have
trajectories which coincide after a specific pair-dependent number of steps.
Then we derive a number of standard total stopping time equalities, including
the ones in Garner (1985), as well as several novel results.