Patterns in Collatz Trajectories

It's my 72nd birthday and I'm 26298 days old. It's a number that seems to have no interesting properties, despite an extensive search. I decided to look at its Collatz trajectory and that turns out to have 77 steps. This led me to consider what other numbers between 1 and 100,000 also have trajectories of this length. Figure 1 shows what I found.

Figure 1: numbers with Collatz trajectories of length 77 steps
in range from 1 to 100,000

The results are interesting. There are an initial three numbers (790, 791, 793) and then a big gap until 4252 is reached. After this there are many numbers with a trajectory of length 77:

4252, 4253, 4254, 4260, 4261, 4262, 4278, 4279, 4289, 4290, 4291, 4295, 4305, 4306, 4307, 4338, 4339, 4350, 4351, 4374, 4375, 4377, 4380, 4381, 4382, 4383, 4409, 4418, 4419, 4423, 4434, 4435, 4494, 4495, 4744, 4746, 4748, 4749, 4752, 4756, 4757, 4758, 4759, 4760, 4762, 4769, 4779, 4783

There is a jump after 4783 to 5279 and 5287 after which there is a very big gap until 25520 is reached, after which there is a steady stream of numbers (with minor gaps) until 31855 is reached. After this, nothing up to the calculation limit of 100,000. There are 439 numbers in total.

Contrast this with numbers that have trajectories of length 76 steps and 78 steps, with 1417 and 908 numbers respectively. See Figures 2.

Figure 2: numbers with Collatz trajectories of length 76 steps
in range from 1 to 100,000

Figure 3: numbers with Collatz trajectories of length 78 steps
in range from 1 to 100,000

The results for trajectories of length of 76 and 78 steps led me to suspect that there was probably another run of numbers above 100,000 with trajectories of 77 steps. I extended the search up to 200,000 and Figure 4 shows what I found.

Figure 4: numbers with Collatz trajectories of length 77 steps
in range from 1 to 200,000

In the range up to 200,000, there are 2751 numbers with trajectories of 77 steps. Figures 2, 3 and 4 are nearly identical in their essential features, only differing in where the runs begin and end. Dropping down to numbers with trajectories of length 38 steps, the same pattern recurs. See Figure 5.

Figure 5: numbers with Collatz trajectories of length 38 steps
in range from 1 to 200,000

I've no idea why these patterns occur and am simply noting them in this post. Thus 26298 turns out to be a number of some interest after all because it led me to discover this pattern. As can be seen from Figure 1 and Figure 4, there are many numbers in the vicinity of 26298 that also have trajectories of 77 steps:

..., 26292, 26293, 26296, 26298, 26300, 26301, 26302, 26305, ...

Of course, looking at the list above, I'm interested to know what the trajectories are for the numbers between 26292 and 26305 that aren't in the list. Here's what I found.

• 26294 requires 64 steps
• 26295 requires 64 steps
• 26297 requires 64 steps
• 26299 requires 64 steps
• 26303 requires 64 steps
• 26304 requires 139 steps

Let's keep this going for a bit longer:

• 26306 requires 100 steps
• 26307 requires 100 steps
• 26308 requires 139 steps
• 26309 requires 139 steps
• 26310 requires 139 steps

Obviously, this is just a small sample of the behaviour of the non-77 step numbers surrounding 26298. More analysis is needed before any generalisations can be made but oddly, whether the number is odd or even, it doesn't seem to make much difference. This is in spite of the fact that a number like 26294 --> 13147 while 26295 --> 78886. However, if go another step see that 13147 --> 39442 while 78886 --> 39443 and the numbers are paired again.