I did a more direct test. Using iterate c(n) = n/2 if n even; (3n+1)/2 if n odd I looked at small b >= 1, counting the trajectories of n on [0, 2^b), that first reached a value < n on the b-th step. It turns out that for certain b, the trajectory of n never reaches a value < n for the first time on the b-th step. My program says these b are (3, 6, 9, 11, 14, 17, 19, 22, 25, 28, 30, .). If my interpretation of A182137 is correct, these are the b for which A182137(b) = 2*A182137(b-1). These b are (3, 6, 9, 11, 14, 17, 19, 22, 25, 28, 30, 33, 36, 37, .). I might have guessed that the b fall into a Beatty-like sequence b(n) = ceil(c*n) with 2.7 <= c < 3. However, in such a sequence, first differences must be either 2 or 3, inconsistent with adjacent elements 36 and 37 implied by A182137. This means either my intuition is wrong (the set of b is more interesting and mysterious than a Beatty sequence), or A182137 is wrong.