Visualizing Prime cyclical data

Question

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!

Answer 1

You can cast your results into a directed graph as follows:

data = primeCycle /@ Prime[Range[100]];

Graph[
 DirectedEdge @@@ Union @@ Map[Partition[#, 2, 1] &, data],
 VertexLabels -> "Name",
 ImageSize -> Large
 ]