The 3n+1 problem

Consider the following function:

f(n)={n/2 if n even, 3n+1 if n odd}, n in N.

We ask what happens when we apply f continuously given a certain natural n.

For example, for n=3 we get the sequence: 3, 10, 5, 16, 8, 4, 2, 1. Will any number eventually stabilize on 1? If you try to program a trivial little program that shows you this sequence, you will be convinced that eventually all tried numbers go to 1.

Given that, define a sequence p_n=period of n. So in the example above, p₃=8. Now define another sequence as follows:

φ_m,n={1≤k≤m: p_k=n}. So essentially, φ_m,n counts how many numbers have a given period n in the range 1≤k≤m.

Does φ_m,n converge? More specifically, does lim_m->∞φ_m,n exist for given n? What about lim_n->∞p_n?

The 3n+1 question or Collatz Conjecture is an unsolved problem. Given that, one would expect that the second question would be unsolved as well. It appears that φ_m,n diverges. The following are graph plots of φ_m,n versus p_n. Successively they are φ_200,n φ_400,n φ_800,n φ_8000,n and φ_20000,n. p_n ranges on the graphs from 1 to 124, 143, 170, 261 and 278: