[seqfan] Re: Patterns of congruence classes modulo 2^n in reduced Collatz sequences

[Bob Selcoe]

Hi Seqfans,

Thanks everyone for your replies.

Max - I will try to post the array as efficiently as possible.  I hope the editors appreciate it! ;-)

Ed and David - thanks for the info, but I think I'm saying something different.  

Ed, a few weeks ago I posted several formulas on https://oeis.org/A075677; they apply to next term in sequence.  The array actually does, too, but differently.  The array also does other things (more in a moment). 

David -  your description is what I mean by "reduced Collatz"; as far as the n mod 2^k idea, I thinks that's also something different.

Perhaps some examples will help clarify.  Treat T(n,k) as the least residue in a congruency class T(n,k) mod 2^(n+k).  The array starts T(2,1).  Members of that class are starting terms S(0) in reduced Collatz sequences; subsequent terms are S(j) j>=1.

The union of the members of all the congruency classes is the unique set of odd natural numbers.

So for instance, T(7,5) = 3935 means the members in congruency class T(n,k) mod 2^(n+k) == 3935 mod 4096 will take k=5 steps to reach the first term < than its preimage (call this term destination (D)), and n=7 halving steps from preimage to D.

(Note: "==" means "congruent to").  

It's very easy to calculate all T(n,k); there is a pattern that's quite regular but yields some random-looking results.  For example, starting with n=2, T(n,6) starts {191, 319, 63, 1599, 575, 2623, 14911, 6719, 55871...}.

Moreover, starting with S(0), it's possible to predict the congruence class of (most) terms S(j) j>=1 by knowing only T(n,k).  By "most", I mean the trajectories are perfectly predictable up until the pattern breaks down; however, the "unpredictability" point (or "bounce" point) itself is fully predictable.  The trajectories go through a cycle of predictable trajectories and "bounce" points, until (presumably) reaching 1 eventually. 

Again take example T(7,5) = 3935: consider S(0) = 8031, which is in class 3935 mod 4096.  Without getting into detail, we can predict using the array that if S(0) = 8031:  S(1) ==  1807 mod 2048; S(2) == 663 mod 1024; S(3) == 483 mod 512; and S(4) == 213 mod 256. 
The bounce point is  213 mod 256.  I can't find a pattern for what happens here; but once empirically we find S(5), we can place it in the array and continue as before, following whatever path is predicted by the array. (In some cases bounce points lead to other bounce points, or 1).  

There are other interesting properties as well.

At the moment all of this is conjecture - I've done this by hand with an online Collatz calculator.  I don't program so the process has been very cumbersome.  But I've randomized some larger results and they have all held up.  I think the properties will prove to hold.  

Anyway, maybe all of these are known properties (are they??), but I think the array is at least interesting enough for an OEIS entry, even stating the properties as conjectures.  

The array read by diagonals starts {1, 13, 11, 5, 19, 7, 53, 3, 55, 47...}   It is not in the OEIS.

Best Wishes, 
Bob



--------------------------------------------------
From: "L. Edson Jeffery" <lejeffery2 at gmail.com>
Sent: Saturday, April 25, 2015 11:17 AM
To: <seqfan at list.seqfan.eu>
Subject: [seqfan] Re: Patterns of congruence classes modulo 2^n in reduced Collatz sequences

> Bob,
> 
> Cf. https://oeis.org/A075677 . (I forgot to mention this.)
> 
> Ed Jeffery
> 
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