Collatz Conjecture, why an increment of $+6$ in the following?

Question

Following my previous question at: Collatz Conjecture, why a rate of change of $*4$ in the following?

Following the rules of the Collatz Conjecture, in this experiment I have created a list of all odd numbers until $33333$. The list includes 3 columns, such as in the following sample:

A) Starting Odd $(X)$	B) $(X * 3) +1$	C) $X/2$ repeat until odd
1	4	2, (1)
3	10	(5) **
5	16	8, 4, 2, (1)
7	22	(11) **
9	28	14, (7)
11	34	(17) **
13	40	20, 10, (5)
15	46	(23) **
17	52	26 (13)
19	58	(29) **

...

You will notice that starting with $X=3$ in column A) and incrementing $X + 4$ in column A) will result in a consistent increment of $+6$ in column C) (marked by ** in the table). Why is that ?

Also the same doesn't apply when starting with $X=1$ in column A) and incrementing $X + 4$ in column A). Why is that?

Answer 1

The lines marked ** are those where the odd number is of the form $4k+3$ When you apply the Collatz iteration the first time you get $3(4k+3)+1=12k+10$. This can only be divided by $2$ once, which is the second iteration. Since the numbers of type $4k+3$ increment by $4$, after one iteration they increment by $12$ and after the second they iterate by $6$. On the other hand, numbers of the form $4k+1$ will go to $12k+4$ and can be divided by $2$ at least twice on the second iteration.