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,
p_{3}=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:

More on this on MathWorld.