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:

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

Does ö_{m,n} converge? More specifically, does lim_{m->∞}ö_{m,n}
exist for given n? What about lim_{n->∞}p_{n}?

It turns out the 3n+1 question is an unsolved problem. Given that, one
would expect that the second question would be unsolved as well. It appears
that ö_{m,n} diverges. The following are graph plots of ö_{m,n}
versus p_{n}. Successively they are ö_{200,n }ö_{400,n} ö_{800,n }ö_{8000,n}
and ö_{20000,n}. p_{n} ranges on the graphs from 1 to 124, 143, 170, 261 and 278:

Many of these questions are investigated on mensanator's pages.