A Special Class of Collatz Trajectories

I've written about Collatz trajectories over the years but a special class of trajectory was brought to my attention thanks to the number, 27074, associated with my diurnal age today. This number is a member of OEIS A224303:

A224303

Numbers \(n\) for which number of iterations to reach the largest equals number of iterations to reach 1 from the largest in Collatz (3\(x\)+1) trajectory of \(n\).

The trajectory of 27074 requires 64 steps to reach 1 and therefore 65 numbers are involved once we include the starting number. These are the numbers (permalink):

27074, 13537, 40612, 20306, 10153, 30460, 15230, 7615, 22846, 11423, 34270, 17135, 51406, 25703, 77110, 38555, 115666, 57833, 173500, 86750, 43375, 130126, 65063, 195190, 97595, 292786, 146393, 439180, 219590, 109795, 329386, 164693, 494080, 247040, 123520, 61760, 30880, 15440, 7720, 3860, 1930, 965, 2896, 1448, 724, 362, 181, 544, 272, 136, 68, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1

The maximum value reached is 494080 and this is the 33rd term. There are 32 terms before it and 32 terms after it. In other words, 32 steps are required to reach the maximum and from there 32 steps are required to reach 1. This is shown in Figure 1:

Figure 1: permalink

The terms in OEIS A224303 up to 40000 are as follows (permalink):

1, 6, 120, 334, 335, 804, 1249, 2008, 2010, 2012, 2013, 6556, 6557, 6558, 6801, 6802, 6803, 7496, 7498, 7500, 7501, 7505, 10219, 22633, 25182, 25183, 27074, 27075, 27864, 27866, 27868, 31838, 31839, 32078, 36630, 36633, 36690, 36691, 36914, 39126, 39344, 39346, 39348, 39352, 39354

The terms appear in clusters. For example, 27075 belongs to the sequence as well as 27074. This clustering is more evident once we plot the numbers as shown in Figure 2:

Figure 2: permalink

It can be noted that the Collatz trajectory of 27075 also requires 64 steps to reach 1, attains the same maximum value of 494080 after 32 steps and then another 32 steps to reach 1.  The trajectory is as follows:

27075, 81226, 40613, 121840, 60920, 30460, 15230, 7615, 22846, 11423, 34270, 17135, 51406, 25703, 77110, 38555, 115666, 57833, 173500, 86750, 43375, 130126, 65063, 195190, 97595, 292786, 146393, 439180, 219590, 109795, 329386, 164693, 494080, 247040, 123520, 61760, 30880, 15440, 7720, 3860, 1930, 965, 2896, 1448, 724, 362, 181, 544, 272, 136, 68, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1

Compare this to that of 27074, we see that they differ only in first five terms before merging at 30460:

27074, 13537, 40612, 20306, 10153, 30460, 15230, 7615, 22846, 11423, 34270, 17135, 51406, 25703, 77110, 38555, 115666, 57833, 173500, 86750, 43375, 130126, 65063, 195190, 97595, 292786, 146393, 439180, 219590, 109795, 329386, 164693, 494080, 247040, 123520, 61760, 30880, 15440, 7720, 3860, 1930, 965, 2896, 1448, 724, 362, 181, 544, 272, 136, 68, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1

So overall an interesting class of Collatz trajectories, just adding to the fascination of the 3\(x\)+1 problem.