More steps

In[]:=
Graph[ResourceFunction["NestGraphTagged"][n|->Table[an,{a,4}],1,6],VertexLabels->Automatic,GraphLayout"LayeredDigraphEmbedding"]
Out[]=

More multipliers

In[]:=
Graph[ResourceFunction["NestGraphTagged"][n|->Table[an,{a,5}],1,4],VertexLabels->Automatic,GraphLayout"LayeredDigraphEmbedding"]
Out[]=

More initial conditions

In[]:=
Graph[ResourceFunction["NestGraphTagged"][n|->Table[an,{a,4}],Range[5],4],VertexLabels->Automatic,GraphLayout"LayeredDigraphEmbedding"]
Out[]=

Is every number connected to every other number?

No, because it’s only connected to multiples of itself

Common ancestor between two numbers

Given a number, its path is defined by a sequence of divisors of the number
In[]:=
Divisors[12]
Out[]=
{1,2,3,4,6,12}
In[]:=
IntegerPartitions[7]
Out[]=
{{7},{6,1},{5,2},{5,1,1},{4,3},{4,2,1},{4,1,1,1},{3,3,1},{3,2,2},{3,2,1,1},{3,1,1,1,1},{2,2,2,1},{2,2,1,1,1},{2,1,1,1,1,1},{1,1,1,1,1,1,1}}
In[]:=
MultiplicativeParts[n_]:=Flatten[Table[##]&@@@FactorInteger[n]]
In[]:=
AllMultiplierDecompositions[n_]:=Union[With[{m=MultiplicativeParts[n]},Catenate[Function[u,(Times@@@TakeList[u,#])&/@Catenate[Permutations/@IntegerPartitions[Length[m]]]]/@Permutations[m]]]]
In[]:=
AllMultiplierDecompositions[12]
Out[]=
{{12},{2,6},{3,4},{4,3},{6,2},{2,2,3},{2,3,2},{3,2,2}}
In[]:=
AllMultiplierDecompositions[13]
Out[]=
{{13}}
In[]:=
AllMultiplierDecompositions[14]
Out[]=
{{14},{2,7},{7,2}}
In[]:=
AllMultiplierDecompositions[20]
Out[]=
{{20},{2,10},{4,5},{5,4},{10,2},{2,2,5},{2,5,2},{5,2,2}}
In[]:=
Table[AllMultiplierDecompositions[n],{n,20}]
Out[]=
{{{1}},{{2}},{{3}},{{4},{2,2}},{{5}},{{6},{2,3},{3,2}},{{7}},{{8},{2,4},{4,2},{2,2,2}},{{9},{3,3}},{{10},{2,5},{5,2}},{{11}},{{12},{2,6},{3,4},{4,3},{6,2},{2,2,3},{2,3,2},{3,2,2}},{{13}},{{14},{2,7},{7,2}},{{15},{3,5},{5,3}},{{16},{2,8},{4,4},{8,2},{2,2,4},{2,4,2},{4,2,2},{2,2,2,2}},{{17}},{{18},{2,9},{3,6},{6,3},{9,2},{2,3,3},{3,2,3},{3,3,2}},{{19}},{{20},{2,10},{4,5},{5,4},{10,2},{2,2,5},{2,5,2},{5,2,2}}}
In[]:=
AllMultiplierDecompositions[12]
Out[]=
{{12},{2,6},{3,4},{4,3},{6,2},{2,2,3},{2,3,2},{3,2,2}}
In[]:=
AllMultiplierDecompositions[20]
Out[]=
{{20},{2,10},{4,5},{5,4},{10,2},{2,2,5},{2,5,2},{5,2,2}}
In[]:=
AllMultiplierDecompositions[18]
Out[]=
{{18},{2,9},{3,6},{6,3},{9,2},{2,3,3},{3,2,3},{3,3,2}}
For any sequence of exponents of primes, the structure of the decomposition will be the same...
In[]:=
FactorInteger[20]
Out[]=
{{2,2},{5,1}}
In[]:=
FactorInteger[12]
Out[]=
{{2,2},{3,1}}

Number of paths of a given length to a given number

In[]:=
Table[KeySort[Counts[Length/@AllMultiplierDecompositions[n]]],{n,20}]
Out[]=
{11,11,11,11,21,11,11,22,11,11,22,31,11,21,11,22,11,11,24,33,11,11,22,11,22,11,23,33,41,11,11,24,33,11,11,24,33}
For primes, there’s only one path (and it’s trivial, because they must be there at the beginning)
In[]:=
KeySort[Counts[Length/@AllMultiplierDecompositions[12]]]
Out[]=
11,24,33
In[]:=
KeySort[Counts[Length/@AllMultiplierDecompositions[18]]]
Out[]=
11,24,33
In[]:=
KeySort[Counts[Length/@AllMultiplierDecompositions[2^5]]]
Out[]=
11,24,36,44,51
In[]:=
KeySort[Counts[Length/@AllMultiplierDecompositions[2^6]]]
Out[]=
11,25,310,410,55,61
What this shows is how frequently the number will show up at a given step number (i.e. its path weight at that step number)
Product of 4 primes
For a number n to be reached from an initial condition i, i must be a divisor of n ... and the “remaining path” must be the path for the number n/i

From a given number, consider its light cone ...

If one has already taken the multiplicand limit, the light cone has infinite opening angle
E.g. it’s not possible to nontrivially reach a prime after 2 steps

Can one find a composition of graphs that reflects the composition of numbers?

The graph is “uniform” i.e. vertex transitive.... for outgoing edges [but not incoming]
When there are distinct prime factors, the graph one gets is a s-dimensional grid when there are s prime factors

How many distinct numbers can I first reach (with m multiplicands) after t steps?

Exponent graphs