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thegreatjared

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For example, take 28. Its prime factors (taken with multiplicity) are 2, 2, and 7. The sum of these is 11. Double it and you get 22. So, the sum of prime factors of 28 multiplied by 2 is 22. Repeat this process until you reach a loop or a stable number.

For 28, if you carry this process out to its conclusion it yields 28-->22-->26-->30-->20-->18-->16-->16-->…

As you can see, when you reach 16, the process remains "stuck." My question is this: Will any number, n, always reach 16 in this process, or are there perhaps loops that the process could reach, as well?

I think it is obvious that there are no other numbers besides 16 that the process could get hung up on, but this doesn't rule out the possibility that there are loops that the process could get stuck on. Although, it is my hunch that this is not the case and that the case of repeated 16s will prove to be the only ending to any number, n, in this process.