Numerical Ruliad...

In[]:=
gg=Table[Last[ResourceFunction["GraphNeighborhoodVolumes"][With[{m=k},NestGraph[n|->Table[an,{a,m}],1,10]],{1}]],{k,2,13}]
Out[]=
{{1,2,3,4,5,6,7,8,9,10,11},{1,3,6,10,15,21,28,36,45,55,66},{1,4,9,16,25,36,49,64,81,100,121},{1,5,14,30,55,91,140,204,285,385,506},{1,6,18,40,75,126,196,288,405,550,726},{1,7,25,65,140,266,462,750,1155,1705,2431},{1,8,30,80,175,336,588,960,1485,2200,3146},{1,9,36,100,225,441,784,1296,2025,3025,4356},{1,10,42,120,275,546,980,1632,2565,3850,5566},{1,11,53,173,448,994,1974,3606,6171,10021,15587},{1,12,59,194,504,1120,2226,4068,6963,11308,17589},{1,13,72,266,770,1890,4116,8184,15147,26455,44044}}
In[]:=
ff=Table[LinearRecurrence[Table[(-1)^(k+1)Binomial[PrimePi[m]+1,k],{k,1,PrimePi[m]+1}],Prepend[Length/@NestList[Union[Times@@#&/@Tuples[{Range[m],#}]]&,Range[m],PrimePi[m]-1],1],11],{m,2,13}]
Out[]=
{{1,2,3,4,5,6,7,8,9,10,11},{1,3,6,10,15,21,28,36,45,55,66},{1,4,9,16,25,36,49,64,81,100,121},{1,5,14,30,55,91,140,204,285,385,506},{1,6,18,40,75,126,196,288,405,550,726},{1,7,25,65,140,266,462,750,1155,1705,2431},{1,8,30,80,175,336,588,960,1485,2200,3146},{1,9,36,100,225,441,784,1296,2025,3025,4356},{1,10,42,120,275,546,980,1632,2565,3850,5566},{1,11,53,173,448,994,1974,3606,6171,10021,15587},{1,12,59,194,504,1120,2226,4068,6963,11308,17589},{1,13,72,266,770,1890,4116,8184,15147,26455,44044}}
In[]:=
Table[Prepend[Length/@NestList[Union[Times@@#&/@Tuples[{Range[m],#}]]&,Range[m],PrimePi[m]-1],1],{m,6}]
NestList
:Non-negative machine-sized integer expected at position 3 in NestList[Union[(Times@@#1&)/@Tuples[{Range[m],#1}]]&,{1},-1].
Out[]=
{{1,1},{1,2},{1,3,6},{1,4,9},{1,5,14,30},{1,6,18,40}}
In[]:=
Table[FindSequenceFunction[LinearRecurrence[Table[(-1)^(k+1)Binomial[PrimePi[m]+1,k],{k,1,PrimePi[m]+1}],Prepend[Length/@NestList[Union[Times@@#&/@Tuples[{Range[m],#}]]&,Range[m],PrimePi[m]-1],1],15],t],{m,2,20}]
Out[]=
t,
1
2
t(1+t),
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t
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1
6
(t+3
2
t
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3
t
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1
2
(
2
t
+
3
t
),
1
24
(2t+9
2
t
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3
t
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4
t
),
1
6
(2
2
t
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3
t
+
4
t
),
1
4
(
2
t
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3
t
+
4
t
),
1
6
(
2
t
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3
t
+2
4
t
),
1
120
(2t+25
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t
+50
3
t
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4
t
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5
t
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1
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(-4t+20
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t
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4
t
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5
t
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1
240
(42
2
t
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t
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5
t
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6
t
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1
720
(-12t+104
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t
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3
t
+245
4
t
+87
5
t
+11
6
t
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1
360
(-6t+43
2
t
+135
3
t
+130
4
t
+51
5
t
+7
6
t
),
1
360
(-12t+32
2
t
+135
3
t
+140
4
t
+57
5
t
+8
6
t
),
-72t+546
2
t
+1729
3
t
+1785
4
t
+847
5
t
+189
6
t
+16
7
t
5040
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1
840
(-24t+70
2
t
+287
3
t
+315
4
t
+154
5
t
+35
6
t
+3
7
t
),
-240t+1972
2
t
+6384
3
t
+7021
4
t
+3780
5
t
+1078
6
t
+156
7
t
+9
8
t
20160
,
-240t+772
2
t
+3164
3
t
+3675
4
t
+2030
5
t
+588
6
t
+86
7
t
+5
8
t
10080

In[]:=
CoefficientList[#,t]&/@%168
Out[]=
{0,1},0,
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,{0,0,1},0,
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,0,-
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2016

In[]:=
Exponent[#,t]&/@%168
Out[]=
{1,2,2,3,3,4,4,4,4,5,5,6,6,6,6,7,7,8,8}
In[]:=
ListStepPlot[%]
Out[]=
[major slowdown around 29 ... }
In[]:=
Monitor[Table[Exponent[FindSequenceFunction[LinearRecurrence[Table[(-1)^(k+1)Binomial[PrimePi[m]+1,k],{k,1,PrimePi[m]+1}],Prepend[Length/@NestList[Union[Times@@#&/@Tuples[{Range[m],#}]]&,Range[m],PrimePi[m]-1],1],15],t],t],{m,2,30}],m]
Out[]=
{1,2,2,3,3,4,4,4,4,5,5,6,6,6,6,7,7,8,8,8,8,9,9,9,9,9,9,10,10}
In[]:=
Differences[%]
Out[]=
{1,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,0,0,0,0,1,0}
In[]:=
ListStepPlot[%175]
Out[]=
In[]:=
Position[%176,1]//Flatten
Out[]=
{1,3,5,9,11,15,17,21,27}
In[]:=
FindSequenceFunction[%,t]
Out[]=
FindSequenceFunction[{1,3,5,9,11,15,17,21,27},t]

Primes

In[]:=
Graph[ResourceFunction["NestGraphTagged"][n|->Select[Table[Prime[a]n,{a,10}],#<100&],1,10,"StateLabeling"->True,"RuleStyling"->None],GraphLayout"LayeredDigraphEmbedding",AspectRatio1/2]
Out[]=