This is related to the Collatz conjecture.
I can't help but feel that these numbers are building something. The way that they move seems to suggest a deeper structure building up underneath them which allows them to fall back to 1. If I could only see that structure. I've tried visualizing these values in so many ways but I haven't yet found their true form, the one that causes the pattern to fall out like an elementary truth. Maybe it doesn't exist, maybe it's just noise. But the problem seems too simply stated for that.
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