How much relation is there between Collatz numbers and well-founded numbers? And do the generated images look similar?
Somewhere in the list of Collatz papers (link in previous post) is a directed graph that doesn't look anything like my graph. It has many vertical columns that eventually join the 4-2-1 loop at the bottom. It also contains all the integers where mine only contains primes and the least-odd-prime-factors of the even numbers.
I have replaced the 4-2-1 loop with the prime 5 as the terminator and I let Mathematica determine the positions of the primes for the graph.
A few examples:
When x=7, we get the sequence {7, 11, 17, 13}. We terminate when the result is less than the starting x. For all primes that end in 3, the next result will be 5 and terminate. For all other primes, the sequence terminates because the lower primes have already been determined to terminate. (i.e., when x=7, we already know that 5 terminated.)
We do repeat some sequences. When x=11, we don't know that it has already terminated because we are only testing termination of the first prime in a sequence. We get the sequence {11, 17, 13}. Then {13}. And then {17}, which terminates because the result (13) is less than 17 and we already know it terminates.
The longest sequence is 8. (Examined to 500,000 primes) There is only one well-ordered sequence of length 8:
{44102911, 66154367, 99231551, 148847327,
223270991, 334906487, 502359731, 753539597}
The others' orders will vary, but will all be greater than the first prime of each sequence.
So, my goal is to find a superior limit that is less than the first prime of each sequence. This is backwards! However, because the function
lopf(3p+1) is a reducing function, I hope it is doable.
It required 1.5 pots of coffee to write this post!
EDIT: I have no idea if it is well-founded.
