In[]:=
Manipulate[With[{gg=NestGraph[n{n+7,n+11,n+13},0,14,VertexLabels->"Name"]},Graph[gg,VertexCoordinates->Thread[VertexList[gg]->(QuotientRemainder[#,divisor]&/@VertexList[gg])],ImageSize->{700,200}]],{{divisor,7},4,11,1,Appearance->"Labeled"}]
Out[]=
​
divisor
7
In[]:=
Graph3D[NestGraph[n{n+7,n+11,n+13},0,20]]
Out[]=
In[]:=
Graph3D[NestGraph[n{n+7,n+11,n+13},0,8]]
Out[]=
In[]:=
With[{g=ResourceFunction["MultiwayFunctionSystem"][n{n+4,n+7,n+9},0,8,"StatesGraphStructure"]},Graph3D[g,EdgeStyle->(#->Switch[ToExpression@Last[#]-ToExpression@First[#],4,Directive[Green,Thick],7,Directive[Blue,Thick],9,Directive[Red,Thick]]&/@EdgeList[g])]]
Out[]=
In[]:=
Table[{Max[FrobeniusNumber[{a,b}],0],CoprimeQ[a,b]},{a,10},{b,10}]
Out[]=
{{{0,True},{0,True},{0,True},{0,True},{0,True},{0,True},{0,True},{0,True},{0,True},{0,True}},{{0,True},{∞,False},{1,True},{∞,False},{3,True},{∞,False},{5,True},{∞,False},{7,True},{∞,False}},{{0,True},{1,True},{∞,False},{5,True},{7,True},{∞,False},{11,True},{13,True},{∞,False},{17,True}},{{0,True},{∞,False},{5,True},{∞,False},{11,True},{∞,False},{17,True},{∞,False},{23,True},{∞,False}},{{0,True},{3,True},{7,True},{11,True},{∞,False},{19,True},{23,True},{27,True},{31,True},{∞,False}},{{0,True},{∞,False},{∞,False},{∞,False},{19,True},{∞,False},{29,True},{∞,False},{∞,False},{∞,False}},{{0,True},{5,True},{11,True},{17,True},{23,True},{29,True},{∞,False},{41,True},{47,True},{53,True}},{{0,True},{∞,False},{13,True},{∞,False},{27,True},{∞,False},{41,True},{∞,False},{55,True},{∞,False}},{{0,True},{7,True},{∞,False},{23,True},{31,True},{∞,False},{47,True},{55,True},{∞,False},{71,True}},{{0,True},{∞,False},{17,True},{∞,False},{∞,False},{∞,False},{53,True},{∞,False},{71,True},{∞,False}}}
In[]:=
Cases[Catenate[%],{_,False}]
Out[]=
{{∞,False},{∞,False},{∞,False},{∞,False},{∞,False},{∞,False},{∞,False},{∞,False},{∞,False},{∞,False},{∞,False},{∞,False},{∞,False},{∞,False},{∞,False},{∞,False},{∞,False},{∞,False},{∞,False},{∞,False},{∞,False},{∞,False},{∞,False},{∞,False},{∞,False},{∞,False},{∞,False},{∞,False},{∞,False},{∞,False},{∞,False},{∞,False},{∞,False},{∞,False},{∞,False},{∞,False},{∞,False}}
In[]:=
Cases[Catenate[%%],{_,True}]
Out[]=
{{0,True},{0,True},{0,True},{0,True},{0,True},{0,True},{0,True},{0,True},{0,True},{0,True},{0,True},{1,True},{3,True},{5,True},{7,True},{0,True},{1,True},{5,True},{7,True},{11,True},{13,True},{17,True},{0,True},{5,True},{11,True},{17,True},{23,True},{0,True},{3,True},{7,True},{11,True},{19,True},{23,True},{27,True},{31,True},{0,True},{19,True},{29,True},{0,True},{5,True},{11,True},{17,True},{23,True},{29,True},{41,True},{47,True},{53,True},{0,True},{13,True},{27,True},{41,True},{55,True},{0,True},{7,True},{23,True},{31,True},{47,True},{55,True},{71,True},{0,True},{17,True},{53,True},{71,True}}
Fix this table...
In[]:=
Text[Grid[Prepend[Table[Prepend[Table[Max[FrobeniusNumber[{a,b}],0],{a,10}],b],{b,5}],Prepend[Range[10],""]],FrameAll,FrameStyle->GrayLevel[.7],Background{None,{GrayLevel[0.9]}}]]
Out[]=
1
2
3
4
5
6
7
8
9
10
1
0
0
0
0
0
0
0
0
0
0
2
0
∞
1
∞
3
∞
5
∞
7
∞
3
0
1
∞
5
7
∞
11
13
∞
17
4
0
∞
5
∞
11
∞
17
∞
23
∞
5
0
3
7
11
∞
19
23
27
31
∞
In[]:=
ResourceFunction["MultiwayFunctionSystem"][n{n+4,n+7},0,8]
Out[]=
{{0},{4,7},{11,14,8},{12,15,18,21},{16,19,22,25,28},{20,23,26,29,32,35},{24,27,30,33,36,39,42},{28,31,34,37,40,43,46,49},{32,35,38,41,44,47,50,53,56}}
In[]:=
Accumulate[Length/@ResourceFunction["MultiwayFunctionSystem"][n{n+2,n+3},0,8]]
Out[]=
{1,3,6,10,15,21,28,36,45}
In[]:=
FindSequenceFunction[%,t]
Out[]=
1
2
t(1+t)
In[]:=
Length/@%
Out[]=
{1,2,3,4,5,6,7,8,9}
n{n+4,n+7}
In[]:=
Accumulate[Length/@ResourceFunction["MultiwayFunctionSystem"][n{n+4,n+7},0,20]]
Out[]=
{1,3,6,10,15,21,28,36,45,55,66,78,91,105,120,136,153,171,190,210,231}
In[]:=
FindSequenceFunction[%,t]
Out[]=
1
2
t(1+t)
In[]:=
Accumulate[Length/@ResourceFunction["MultiwayFunctionSystem"][n{n+1,n+2},0,20]]
Out[]=
{1,3,6,10,15,21,28,36,45,55,66,78,91,105,120,136,153,171,190,210,231}
In[]:=
FindSequenceFunction[%,t]
Out[]=
1
2
t(1+t)
In[]:=
Table[Min[Total/@FrobeniusSolve[{7,11,13},k]],{k,31,150}]
Out[]=
{3,4,3,4,3,4,3,4,3,4,5,4,5,4,5,4,5,4,5,4,5,4,5,6,5,6,5,6,5,6,5,6,5,6,5,6,7,6,7,6,7,6,7,6,7,6,7,6,7,8,7,8,7,8,7,8,7,8,7,8,7,8,9,8,9,8,9,8,9,8,9,8,9,8,9,10,9,10,9,10,9,10,9,10,9,10,9,10,11,10,11,10,11,10,11,10,11,10,11,10,11,12,11,12,11,12,11,12,11,12,11,12,11,12,13,12,13,12,13,12}
In[]:=
ListPlot[%]
Out[]=
20
40
60
80
100
120
2
4
6
8
10
12

Affine case

a n , n + b

Are there identities that start at larger values or only at the origin?

​

For what numbers is there a merging?

The nth merger occurs at position a n^2

step 30: [ did not finish ]

Given the unloose-ended graph g...
Generate branch pairs:
Iteration of u x + 1 gives u^t x +u^t-1
If it’s not “positive density”, the loops get more and more airey....
Estimate of density of numbers:

Comparison

Pairs which connect (?)

[Simpler Case]

Chinese remainder etc.

Should label these with colors
For the finite case of modular arithmetic, the multiway graph is just a finite graph....
[[[ Certain values cannot occur ]]]
0 is not connected here....
[[[ The 0 “leader” was removed by removing loose ends ]]]
Every multiple of 6, only certain numbers can occur.....

Criterion for convergence

Word equation..... : this would be a result valid for any initial condition.....
Starting from x .... seek two words that lead to an identical result.....
Cf Post correspondence problem....

Free Semigroup Claim

Inverse 3n+1 Problem

The Inverse Problem

Since the density is low, there should be numbers you can never reach...

Most numbers don’t have inversions ... but since the numbers always get smaller, from any number there’s a way back to the root....

There’s an upper bound on how long it takes to get to a given number....

3n+2, 2n+1

a n , n+1