Consider the following function:
f(n)={n/2 if n even
, 3n+1 if n odd}, n in N.

Now ask: What happens when you apply f continuously given a certain natural n?

For example, for n=3 we get the sequence: 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:

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

Ask: Does phi_m,n converge? More specifically, does lim_m->oophi_m,n  exist for given n? What about lim_n->oop_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 phi_m,n diverges. The following are graph plots of phi_m,n versus p_n. Successively they are:
phi_200,n phi_400,n phi_800,n phi_8000,n and phi_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.

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