I'm not too familiar with Mathematica graphics, but I would like to produce a "ring-list" with other lists point to a center list. Here is an example list of the many I've generated (each list is infinite, I think):
$$ \left( \begin{array}{c} \{2,5,11,23,47,19,13,3,7\} \\ \{3,7,5,11,23,47,19,13\} \\ \{5,11,23,47,19,13,3,7\} \\ \{7,5,11,23,47,19,13,3\} \\ \{11,23,47,19,13,3,7,5\} \\ \{13,3,7,5,11,23,47,19\} \\ \{17,7,5,11,23,47,19,13,3\} \\ \{19,13,3,7,5,11,23,47\} \\ \{23,47,19,13,3,7,5,11\} \\ \{29,59,17,7,5,11,23,47,19,13,3\} \\ \end{array} \right) $$
*it won't let me put in the LaTeX because it appears as code.
As you can tell, there are duplicates in this list. The first list is the "root" cycle. I would like all of the other lists to point, respectively, to there belonging connection to this "root" list.
Here is the type of thing I'm envisioning:
Pardon my poor paint skills (and lack of knowledge). If you're curious, I'm generating arithmetic prime sequences with the following algorithm:
primeCycle[x_] := Module[{},
cycleList = {};
h = x;
AppendTo[cycleList, h];
h = Last[FactorInteger[2*h + 1]][[1]];
While[! MemberQ[cycleList, h], {AppendTo[cycleList, h], h =
Last[FactorInteger[2*h + 1]][[1]];}];
cycleList
]
I plan on investigating more than $2\cdot h+1$, but I'm not able to compile enough data by hand. The hope is that maybe I learn something interesting.
I believe that for any function $A\cdot h \pm b$, ($A$ is prime and $b<(A-1)/2)$, there is always a "root" cycle (as depicted above).
I also think it may be interesting to investigate other functional forms, but I plan on sticking with simple functions for now.
Thanks!