Position and role of loops in mod 12 and 16

Applying moduli 12 and 16 to the Collatz procedure generates loops that share some characteristics but not others.

A layman's definition of a loop is a partial sequence of n mod x that repeats itself.

Let's start with mod 12, as it is easier (for me) to illustrate, each type of loop being connected with a type of segment, the brackets indicating the loop:

• Yellow: the loop is [4-2-1]; it remains valid in moduli multiples of 12:
• Green; the loop is [10-11]; more generally, it is [x-2, x-1] mod x.
• Blue; the loop is [4-8]; more generally, it is [x/3, 2x/3] mod x.
• Rosa; the loop is [12]; more generally, it is [x] mod x.

So, any infinite rosa segment is of the form [12]-6-3 mod 12 or [12]-6-9 mod 12.

Quite similarly, any series of blue segments is of the form [4-8]. The main difference is that this loop can start and end "in the middle" of a sequence and occur more than once.

Yellow loops occur in series of triplets alternating with pairs (search "Isolation mechanism" here).

Green loops occur in series of convergent pairs (search here).

Now, how does this change in mod 16 ?

The equivalent exists for yelllow loops, still [4-2-1], green ones, now [14-15], and rosa ones, now [16], but not blue ones.

[To be continued.]