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..

  • $\begingroup$ You are going to have to be clearer. $\endgroup$ – Thomas Andrews Jul 20 '16 at 23:15
  • $\begingroup$ Should make sense now. Although I'm not sure why it didn't in the first place? The 4K+1 pattern seems to be well known. I find it hard to believe i'm the first to find these other patterns - I'm an undergrad. $\endgroup$ – Ben Crossley Jul 20 '16 at 23:21
  • $\begingroup$ Sorry, but problems should best be self-contained, and you didn't say what the pattern was. If you want the most help here, it is best to be as self-contained as possible. $\endgroup$ – Thomas Andrews Jul 20 '16 at 23:44
  • $\begingroup$ Cheers Thomas. I'm used to forums where people don't like to read. I will make sure of that in the future! $\endgroup$ – Ben Crossley Jul 20 '16 at 23:53
  • $\begingroup$ In go.helms-net.de/math/collatz/aboutloop/collloopintro_main.htm on the page of "about numerical and graphical trees" I've examples for the 64k pattern; that all cumulated in the "bottle-brush-tree" which generalizes that stepwidths in a -hopefully- interesting view. $\endgroup$ – Gottfried Helms Jul 21 '16 at 10:16

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".

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