@frank introduced me to the Collatz conjecture yesterday and I found it very interesting. I'm not a mathematician by any means but I thought it would be fun to look into it, just as an exercise. I'd appreciate feedback from any mathematicians as to whether I'm taking the right approach.

This isn't an attempt at a proof. Rather it's an attempt to reformulate the conjecture.

I started at the end and worked backwards through the logic. If the conjecture is true then for all positive integers $n$, and where $m,p,q,...$ are some natural number, one of these equations is true:

$n=1\\n=2^m\\3n+1=2^m\\3{n\over2^p}+1=2^m\\3{3n+1\over2^p}+1=2^m\\3{3{n\over2^q}+1\over2^p}+1=2^m\\...$

The first thing I noted is that the first equation is equivalent to the second equation where $m=0$, that the third equation is equivalent to the fourth equation where $p=0$, and so on down the list, and as such are superfluous and can be removed from consideration (these removed lines apply only to odd-numbers, the remaining lines to both odd- and even-numbers).

I then solved each remaining equation for $n$:

$n=2^m\\n={2^p(2^m-1)\over3}\\n={2^q2^p(2^m-1)\over9}-{2^q\over3}\\n={2^r2^q2^p(2^m-1)\over27}-{2^r2^q\over9}-{2^r\over3}\\...$

Given that $2^r2^q2^p=2^{r+q+p}$ I decided to simplify such occurrences to $2^a$. I also decided to multiply both sides of the equation to remove the fractions:

$n=2^a\\3n=2^a-2^b\\9n=2^a-2^b-(3\cdot2^c)\\27n=2^a-2^b-(3\cdot2^c)-(9\cdot2^d)\\...$

The equations show a strict pattern which I decided to make more explicit:

$3^0\cdot{n}=2^a\\3^1\cdot{n}=2^a-(3^0\cdot2^b)\\3^2\cdot{n}=2^a-(3^0\cdot2^b)-(3^1\cdot2^c)\\3^3\cdot{n}=2^a-(3^0\cdot2^b)-(3^1\cdot2^c)-(3^2\cdot2^d)\\...$

If the conjecture is true then for each positive integer $n$ one of those equations is satisfied.

At this point I am confident in my reasoning. However, what I would like to do is express the above in a more succint manner. It is here that I am less certain. I believe that I can do so with sigma notation, and so this is my attempt to reformulate the Collatz conjecture:

$\forall{n}\in\mathbb{N}^+\exists{x}\in\mathbb{N}:n={2^{a_x}+\sum\limits_{y=0}^{x-1}-(3^y\cdot2^{a_y})\over3^x}$

So, is my notation correct? And if so is it a useful step in trying to prove (or disprove) the conjecture? Or has this been a complete waste of time?

(Side question: is there a better way to symbolize variable powers of 2 than $2^{a_x} ... 2^{a_y}$?)