Questions about n < 2^k

[mathkook]

@AcidicJello

Since I’m here asking questions, I’ve heard a lot of different claims about the maximum value of β for a cycle. We all know the minimum is 2^k - 3^x, but finding the upper bound is a lot harder. If it’s derived from the “high cycle”, where every division by two that keeps the current n >= the initial n, then it would be nice to have a closed form we know will always be larger, but not too much larger, than the definitely not closed form of the high cycle.

For \beta values of rational cycles, I use this picture. There’s a formula for the smallest \beta of all cycles’ members (211 = 3^5 - 2^5), and also for the largest (1688). There’s no formula for the high cycle’s smallest member’s \beta = 319, or its largest \beta = 842, though these quantities can be pretty tightly bounded in terms of k and x.

Looking only at (putative) integer cycles, instead of rational cycles, involves different stuff.