Collatz Variations

Collatz Variations

Consider the mapping:

N  --->  N/2      if N is even

N  --->  3N+1   if N is odd

When iterated, does every number end in the cycle 1--> 4 --> 2 --> 1 ? That is the Collatz Conjecture.

Answer: No, if you include non-positive numbers.

0  -->  0

-1 --> -2 --> -1

-5 --> -14 --> -7 --> -20 --> -10 --> -5

-17 --> -50 --> -25 --> -74 --> -37 --> -110 --> -55 --> -164 --> -82 --> -41 --> -122 --> -61 --> -182 --> -91 --> -272 --> -136 --> -68  --> -34 --> -17

and so on. Are there infinitely many cycles?

Now consider this variant of the Collatz function:

N  --->  N/2      if N is even

N  --->  3(N+1)   if N is odd

First add one, then times three!

6 --> 3 --> 12 --> 6

16 --> 8 --> 4 --> 2 --> 1 --> 6

20 --> 10 -->5 -->18 --> 9 --> 30 --> 15 --> 48 --> 24  --> 12  --> 6

Does it always end with 6-->3-->12-->6 ?   No:

0 --> 0

-3 --> -6 --> -3

-30 --> -15 --> -42 --> -21 --> -60 --> -30

Are there infinitely many cycles?