More steps
More steps
In[]:=
Graph[ResourceFunction["NestGraphTagged"][n|->Table[an,{a,4}],1,6],VertexLabels->Automatic,GraphLayout"LayeredDigraphEmbedding"]
Out[]=
More multipliers
More multipliers
In[]:=
Graph[ResourceFunction["NestGraphTagged"][n|->Table[an,{a,5}],1,4],VertexLabels->Automatic,GraphLayout"LayeredDigraphEmbedding"]
Out[]=
More initial conditions
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?
Is every number connected to every other number?
No, because it’s only connected to multiples of itself
Common ancestor between two numbers
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
Number of paths of a given length to a given number
In[]:=
Table[KeySort[Counts[Length/@AllMultiplierDecompositions[n]]],{n,20}]
Out[]=
{11,11,11,11,21,11,11,22,11,11,22,31,11,21,11,22,11,11,24,33,11,11,22,11,22,11,23,33,41,11,11,24,33,11,11,24,33}
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[]=
11,24,33
In[]:=
KeySort[Counts[Length/@AllMultiplierDecompositions[18]]]
Out[]=
11,24,33
In[]:=
KeySort[Counts[Length/@AllMultiplierDecompositions[2^5]]]
Out[]=
11,24,36,44,51
In[]:=
KeySort[Counts[Length/@AllMultiplierDecompositions[2^6]]]
Out[]=
11,25,310,410,55,61
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 ...
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?
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?
How many distinct numbers can I first reach (with m multiplicands) after t steps?
Exponent graphs
Exponent graphs