Collatz Conjecture states that if x is an element of natural number, then if we

Question

Collatz Conjecture states that if x is an element of natural number, then if we repeatedly apply the function c to x, then x will eventually go to 1. The Collatz function is the following:

Any number of the form 2^n eventually goes to 1.

Any number of the form (2^n - 1) / 3 also eventually goes to 1.

Is there other infinite families of numbers, like above two families, that eventually go to 1.

You do not need to write the proof. You can just give the families. I will try to prove it myself.

Explanation / Answer

If we analyse the kind of function which has been used for Collatz Conjecture, it is evident to find other infinite families of numbers that can be proven to converge to 1.

Let M: Z-> Z be the function defined by

M(x) = p*x + q if x is odd and p,q are such that p*x +q is odd, and x/2 if x is even.

Thus, if we repeatedly apply the function M to any positive integer, eventually we reach the number 1.

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