Obviously encountering a power of 2 in a Collatz sequence is important as this is what automatically causes the sequence to terminate. If you calculate the first power of 2 that appears in the Collatz sequence for the first 5,000,000 seeds, 93.8% of the time it's 16. The only way for this to happen is if the number in the sequence immediately prior to that is 5 (or if the seed itself is 16). Why do the vast majority of sequences end up at 5? Is it possible to prove that as you test more and more seeds this percentage approaches 100? Maybe! Gonna take a crack at it.

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I understand why only even powers of 2 are ever encountered first by Collatz sequences where the seeds aren't themselves odd powers of 2. It's easy to show that (2\^n-1)/3 is an integer iff n is even. But why 16 all the time? 64 pulls no weight at all and 1024 and 256 are basically on par with each other so it's not like there's some sensible distribution to suggest that the lower the power the more "attractive" it is.

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If I had to guess without analyzing further, I'd say it's because the lowest seed whose sequence can first encounter the powers of 2 rail somewhere higher than 16 is 21 which has the sequence (21, *64*, 32, 16, 8, 4, 2,1). So, given a random seed, if its sequence ever gets below 21, it's going to end up at (..., 5, 16, 8, 4, 2, 1) and there are a lot of ways for a sequence dive below 21, including the 2^k *3 rail, the 2^k *5 rail, 2^k *7, and so on, as well as other paths that don't involve hitting any rail. Even so, I doubt that accounts for the full effect.

Would love to hear anyone's thoughts on this!