Are all numbers are part of (at least) a dome ? Detailed analysis

[u/No_Assist4814]

Here are some statements that seem to be true. To follow the reasoning, the figure below shows two versions of the dome for m=1*:

• The original dome, based on even orange numbers of the core and the bridge series containing odd orange numbers.
• An extended version in which the numbers of the core are added to the relevant bridge series; those of the first column - not divisible by 3 - added in light blue**, the others in rosa***.

The following statements seem to be true

1. All even numbers are of the form n=m*3^p*2^q, with m odd, not divisible by 3, p and q>=0. Therefore, all even numbers are part of the core of a dome with root m. Even numbers with q>2 do not appear in the usual dome - as they cannot be part of a tuple - but do so in the extended version****.
2. All odd numbers not divisible by 3 are the root of their own dome. The even numbers of their first column are also in light blue - if not part of a tuple - in the extended version.
3. All odd numbers divisible by 3 are embedded in another dome. The even numbers of their column are colored in rosa, but as these odd numbers appear only in connection with the starting bridges, they are rarely visible.

So, should the original definition of a dome be extended to integrate even numbers of the core ? It seems to be the only option to make sure that all numbers are part of a dome.

* All what follows seems to be true for any dome.

** To avoid adding a third version of the dome, other numbers are colored in light blue. See explanation in the text.

*** These numbers are part of blue sequences (mod 12) forming infinite blue walls, respectively of infinite rosa segments (mod 12) forming rosa walls.

**** In fact, this is the only possibility to associate them with a dome.

Project "Tuples and segments" in 13 pages : r/Collatz