I have seen documentation on the $4K+1$ pattern, but as of yet I have seen nothing on the $64K+35$ pattern or the $262144K+184471$ pattern. Is there anywhere I can read up on these?
I created the general pattern too:
$$2^{2×3^{a-1}}K + (1-\frac{2^a}{3^a}) × (2^{2×3^{a-1}}-1)$$
The 4K+1 pattern adds 2 more division steps after the first ×3+1
The 64K+35 pattern adds 6 more division steps after the second ×3+1
The 262144K+184471 pattern adds 18 more division steps after the third ×3+1
and so on..
About the 1+4k pattern: If you study the map T(x) = (1+3x)/2 for x odd and x/2 for x even, then if you iterate that map, starting with an odd number, you get a list of values in increasing order, until just before you reach an even value. That last odd value is of the form 1+4k. You can also show that there is a "maximal list" that ends in this 1+4k. And you can also show that you can partition the positive odd integers using such "lists". You might consider the k in 1+4k to be the "label" of each set in this partition. For each nonnegative integer k, there is exactly one corresponding "maximal list", ending with 1+4k. In each of those "maximal lists", each value (element of the list) is mapped by T into the next, except for the last and largest one, 1+4k. The first few (at least one) T iterate of that 1+4k will be even. The next odd iterate will be either 1+3k, or its largest odd factor, and that will belong in a different "maximal list".
