Things to consider after Collatz conjecture is solved

[Unity+]

Of course it hasn't been solved yet, but it would be significant to consider other properties of Hailstone sequences after it has been solved(or the properties brought in this post will be solved with the solution to the problem).

 

Before introducing these problems, here are things to be defined:

 

• A Collatz number is defined by a number in a Hailstone sequence that is represented by [math]c_{x}=6x-2[/math] where x is the index of the number. This is a result of [math]c_{x}=3(2x-1)+1[/math].
• A Hailstone sequence is a sequence of numbers resulting from the Collatz conjecture rules.
• A Hailstone exception is a number that does not follow the pattern of a Collatz number appearing every other step. For example, {3, 10, 5, 16, 8, 4, 2, 1} does follow this pattern until 4(where 4 is a hailstone exception).
• There is no formula to define Hailstone exception(as of yet that I have developed) because they seem to appear at random points in a Hailstone sequence. However, there might be one I have not seen. There are two types of Hailstone Exceptions(HE). One is the type where it is defined by being [math]c_{x}=2^{n}(n_{o})[/math], where [math]n_{o}[/math] is an odd number. The other is represented by [math]c_{x}=2^{n}(c_{x})[/math].
• EDIT: The formula(I think) of a hailstone exception would be [math]3(4x - 3)+1[/math], or [math]12x - 8[/math]. However, this does not predict if it is one of the two Hailstone exceptions.

Here is an example that contains a few hailstone exceptions(this example contains only Collatz numbers):

{27 82 [124] 94 142 214 322 [484] [364] 274 [412] 310 466 [700] 526 790 1186 1780 (1336) 334 502 754 1132 850 1276 958 1438 2158 3238 4858 7288 1822 2734 4102 6154 9232 4616 1154 1732 1300 976 [244] 184 46 70 106 160 [40] 10 (16) [4]}

 

Here are the issues that might be addressed after the conjecture is solved(unless it is solved within its solution).

 

• For a [math]c_{x}=2^{n}(c_{x})[/math] hailstone exception, what is the largest value of n(that will appear in a hailstone sequence)?
• For a [math]c_{x}=2^{n}(n_{o})[/math] hailstone exception, what is the largest value of n(that will appear in a hailstone sequence)?
• For a hailstone exception, what is the farthest distance between hailstone exceptions?
• What is the formula to represent a hailstone exception? Is there any?

These are just a few things that might be addressed. I find them important from what I have studied within the Collatz conjecture.

 

EDIT2: Just for efficiency, here is the equation for the 2^{x} numbers that are Collatz numbers:

 

[math]c_{s}=6(\sum_{n=1}^{s}\left [ 2^{(2n-1)} \right ]+1)-2=6\sum_{n=1}^{s}\left [ 2^{(2n-1)} \right ]+4[/math]

Edited April 5, 2014 by Unity+