--- Abandoned ---
Reason: I ruled out the extension of the notation beyond the nested array separator level.
I define the left-associative exponentiation-based Hyper-B notation, which generally grows significantly slower than the up-arrow notation (for basic arrays).
The basic array notation uses pretty much the same rules as down-arrow notation.
Format: B(a,b,c,d,...) where all variables (a, b, c, d, ...) are natural numbers (positive integers).
Here # and % are the latter of the notation.
The basic rules are:
B(a) = B(a,1) = a
B(a,b) = a^b
B(#,1) = B(#)
B(a,b,c) = B(B(a,b-1,c),a,c-1)
It is easy to show that B(a,b,c) = avvvv...vvvvb with c down arrows, using down-arrow notation
We introduce the new rules by following:
B(a,b,1,c%) = B(a,a,b,c-1%)
B(a,b,1,1,...,1,1,1,c%) = B(a,a,1,1,...,1,1,b,c-1%)
B(a,b,c,d%) = B(B(a,b-1,c,d%),a,c-1,d%)
We introduce the separators for extended arrays, by the rules by following:
Commas indicate {1} separator.
B(a,b{2}2) = B(a,a,1,1,1,...,1,1,1,2) with b string of 1's
B(a,b{2}c%) = B(a,a,1,1,1,...,1,1,1,2{2}c-1%) with b string of 1's
B(a,b{2}1,c%) = B(a,a{2}a,c-1%)
B(a,b{2}1,1,...,1,1,1,c%) = B(a,a{2}1,1,...,1,1,b,c-1%)
B(a,b{2}1{2}c%) = B(a,a{2}1,1,1,...,1,1,1,2{2}c-1%) with b string of 1's
B(a,b{2}1{2}1{2}...{2}1{2}1{2}c%) = B(a,b{2}1{2}1{2}...{2}1,1,1,...,1,1,1,2{2}c-1%) with b string of 1's
B(a,b{3}c%) = B(a,b{2}1{2}1{2}...{2}1{2}1{2}2{3}c-1%) with b string of 1's separated by {2}
B(a,b{3}1{2}2) = B(a,a{3}1,1,1,...,1,1,1,2) with b string of 1's
B(a,b{n}c%) = B(a,b{n-1}1{n-1}1{n-1}...{n-1}1{n-1}1{n-1}2{n}c-1%) with b rows before the final {n-1} separator
B(a,b{3}1{3}2) = B(a,a{3}1{2}1{2}1{2}...{2}1{2}1{2}2) with b string of 1's separated by {2}
B(a,b{1,2}2) = B(a,a{b}2)
B(a,b{1,c}2) = B(a,a{b,c-1}2)
B(a,b{1,1,1,...,1,1,1,c%}2) = B(a,a{1,1,1,...,1,1,b,c-1%}2)
B(a,b{1{2}2%}2) = B(a,a{1,1,1,...,1,1,1,2%}2) with b string of 1's
B(a,b{1{2}c%}2) = B(a,a{1,1,1,...,1,1,1,2{2}c-1%}2) with b string of 1's
And so on.
The latter of the previous rules remains unchanged. For example, we have a ginormously large number of nested separators, such as B(a,b{1{1{1{1{3}2}2}2}2}2) = B(a,b{1{1{1{1{2}1{2}1{2}...{2}1{2}1{2}2}2}2}2}2) with b string of 1's at the top of the separator layer, and so on.
B(3,4,2)
= B(B(3,3,2),3,1)
= B(B(3,3,2),3)
= B(B(B(3,2,2),3),3)
= B(B(B(B(3,1,2),3),3),3)
= B(B(B(B(3),3),3),3)
= B(B(B(3,3),3),3)
= ((3^3)^3)^3
= (27^3)^3
= 19,683^3
= 7,625,597,484,987
B(2,3,1,3)
= B(2,2,3,2)
= B(B(2,1,3,2),2,2,2)
= B(2,2,2,2)
= B(2,B(2,1,2,2),1,2)
= B(2,2,1,2)
= B(2,2,2)
= B(B(2,1,2),2)
= B(2,2)
= 2^2
= 4
We can easily show that the Left-B notation always returns 4 for the array beyond B(2,n,1,2) per degeneration rule, so that 3 is the smallest base that do not generate array.
B(3,2,1,1,4)
= B(3,3,1,2,3)
= B(3,3,3,1,3)
= B(B(3,2,3,1,3),3,2,1,3)
= B(B(3,3,2,1,3),3,2,1,3)
= B(B(B(3,2,2,1,3),3,1,1,3),3,2,1,3)
= B(B(B(3,3,1,1,3),3,1,1,3),3,2,1,3)
= B(B(B(3,3,1,3,2),3,1,1,3),3,2,1,3)
= B(B(B(3,3,3,2,2),3,1,1,3),3,2,1,3)
= B(B(B(B(3,2,3,2,2),3,2,2,2),3,1,1,3),3,2,1,3)
= B(B(B(B(3,3,2,2,2),3,2,2,2),3,1,1,3),3,2,1,3)
= B(B(B(B(B(3,2,2,2,2),3,1,2,2),3,2,2,2),3,1,1,3),3,2,1,3)
= B(B(B(B(B(3,3,1,2,2),3,1,2,2),3,2,2,2),3,1,1,3),3,2,1,3)
= B(B(B(B(B(3,3,3,1,2),3,1,2,2),3,2,2,2),3,1,1,3),3,2,1,3)
= B(B(B(B(B(B(3,2,3,1,2),3,2,1,2),3,1,2,2),3,2,2,2),3,1,1,3),3,2,1,3)
= B(B(B(B(B(B(3,3,2,1,2),3,2,1,2),3,1,2,2),3,2,2,2),3,1,1,3),3,2,1,3)
= B(B(B(B(B(B(B(3,2,2,1,2),3,1,1,2),3,2,1,2),3,1,2,2),3,2,2,2),3,1,1,3),3,2,1,3)
= B(B(B(B(B(B(B(3,3,1,1,2),3,1,1,2),3,2,1,2),3,1,2,2),3,2,2,2),3,1,1,3),3,2,1,3)
= B(B(B(B(B(B(B(3,3,1,3),3,1,1,2),3,2,1,2),3,1,2,2),3,2,2,2),3,1,1,3),3,2,1,3)
= B(B(B(B(B(B(B(3,3,3,2),3,1,1,2),3,2,1,2),3,1,2,2),3,2,2,2),3,1,1,3),3,2,1,3)
= B(B(B(B(B(B(B(B(3,2,3,2),3,2,2),3,1,1,2),3,2,1,2),3,1,2,2),3,2,2,2),3,1,1,3),3,2,1,3)
= B(B(B(B(B(B(B(B(3,3,2,2),3,2,2),3,1,1,2),3,2,1,2),3,1,2,2),3,2,2,2),3,1,1,3),3,2,1,3)
= B(B(B(B(B(B(B(B(B(3,2,2,2),3,1,2),3,2,2),3,1,1,2),3,2,1,2),3,1,2,2),3,2,2,2),3,1,1,3),3,2,1,3)
= B(B(B(B(B(B(B(B(B(3,3,1,2),3,1,2),3,2,2),3,1,1,2),3,2,1,2),3,1,2,2),3,2,2,2),3,1,1,3),3,2,1,3)
= B(B(B(B(B(B(B(B(B(3,3,3),3,1,2),3,2,2),3,1,1,2),3,2,1,2),3,1,2,2),3,2,2,2),3,1,1,3),3,2,1,3)
= B(B(B(B(B(B(B(B(B(B(3,2,3),3,2),3,1,2),3,2,2),3,1,1,2),3,2,1,2),3,1,2,2),3,2,2,2),3,1,1,3),3,2,1,3)
= B(B(B(B(B(B(B(B(B(B(3,3,2),3,2),3,1,2),3,2,2),3,1,1,2),3,2,1,2),3,1,2,2),3,2,2,2),3,1,1,3),3,2,1,3)
= B(B(B(B(B(B(B(B(B(B(B(3,2,2),3),3,2),3,1,2),3,2,2),3,1,1,2),3,2,1,2),3,1,2,2),3,2,2,2),3,1,1,3),3,2,1,3)
= B(B(B(B(B(B(B(B(B(B(B(3,3),3),3,2),3,1,2),3,2,2),3,1,1,2),3,2,1,2),3,1,2,2),3,2,2,2),3,1,1,3),3,2,1,3)
= B(B(B(B(B(B(B(B(B(B(27,3),3,2),3,1,2),3,2,2),3,1,1,2),3,2,1,2),3,1,2,2),3,2,2,2),3,1,1,3),3,2,1,3)
= B(B(B(B(B(B(B(B(B(19683,3,2),3,1,2),3,2,2),3,1,1,2),3,2,1,2),3,1,2,2),3,2,2,2),3,1,1,3),3,2,1,3)
= B(B(B(B(B(B(B(B(B(B(19683,2,2),19683),3,1,2),3,2,2),3,1,1,2),3,2,1,2),3,1,2,2),3,2,2,2),3,1,1,3),3,2,1,3)
= B(B(B(B(B(B(B(B(B(B(19683,19683),19683),3,1,2),3,2,2),3,1,1,2),3,2,1,2),3,1,2,2),3,2,2,2),3,1,1,3),3,2,1,3)
= B(B(B(B(B(B(B(B(B(19683^19683,19683),3,1,2),3,2,2),3,1,1,2),3,2,1,2),3,1,2,2),3,2,2,2),3,1,1,3),3,2,1,3)
= B(B(B(B(B(B(B(B((19683^19683)^19683,3,1,2),3,2,2),3,1,1,2),3,2,1,2),3,1,2,2),3,2,2,2),3,1,1,3),3,2,1,3)
= B(B(B(B(B(B(B(B(19683^19683^2,3,1,2),3,2,2),3,1,1,2),3,2,1,2),3,1,2,2),3,2,2,2),3,1,1,3),3,2,1,3)
= B(B(B(B(B(B(B(B(3^3^20,3,1,2),3,2,2),3,1,1,2),3,2,1,2),3,1,2,2),3,2,2,2),3,1,1,3),3,2,1,3)
= B(B(B(B(B(B(B(B(3^3486784401,3,1,2),3,2,2),3,1,1,2),3,2,1,2),3,1,2,2),3,2,2,2),3,1,1,3),3,2,1,3)
B(3,4{1,1,3}2)
= B(3,3{1,4,2}2)
= B(3,3{3,3,2}2)
= B(3,3{2,3,2}1{2,3,2}1{2,3,2}2)
= B(3,3{2,3,2}1{2,3,2}1{1,3,2}1{1,3,2}1{1,3,2}2)
= B(3,3{2,3,2}1{2,3,2}1{1,3,2}1{1,3,2}1{3,2,2}2)
Members: 78
Prime-dust = B(3,3) = 3^3 = 27
Prime-grain = B(3,6) = 3^6 = 729
Prime-crystal = B(3,10) = 3^10 = 59,049
Prime-stone = B(3,15) = 3^15 = 14,348,907
Prime-hill = B(3,21) = 3^21 = 10,460,353,203
Prime-mountain = B(3,28) = 3^28 = 22,876,792,454,961
Prime-continent = B(3,36) = 3^36 = 150,094,635,296,999,121
Block-dust = B(4,3) = 4^3 = 2^6 = 64
Block-grain = B(4,6) = 4^6 = 2^12 = 4,096
Block-crystal = B(4,10) = 4^10 = 2^20 = 1,048,576
Block-stone = B(4,15) = 4^15 = 2^30 = 1,073,741,824
Block-hill = B(4,21) = 4^21 = 2^42 = 4,398,046,511,104
Block-mountain = B(4,28) = 4^28 = 2^56 = 72,057,594,037,927,936
Block-continent = B(4,36) = 4^36 = 2^72 = 4,722,366,482,869,645,213,696
Coalstone = B(10,10) = 10^10 = 10,000,000,000
Limonite = B(10,25) = 10^25 = 10,000,000,000,000,000,000,000,000
Calaverite = B(10,50) = 10^50
Koogol / Carbonado = B(10,100) = 10^100
Koogoljex = B(B(10,100),10) = (10^100)^10 = 10^1,000
Koogoldujex = B(B(B(10,100),10),10) = ((10^100)^10)^10 = 10^10,000
Koogoltrijex = B(B(B(B(10,100),10),10),10) = (((10^100)^10)^10)^10 = 10^100,000
Noogol = B(10,100,2) = 10^10^99
Noogoljex = B(B(10,100,2),10,2) = (10^10^99)^(10^10^99)^9
Noogoldujex = B(B(B(10,100,2),10,2),10,2)
Noogoltrijex = B(B(B(B(10,100,2),10,2),10,2),10,2)
Troogol = B(10,100,3) ~ 10^^101
Troogoljex = B(B(10,100,3),10,3)
Choogol = B(10,100,4) ~ 10^^9,901
Panoogol = B(10,100,5) ~ 10^^^101
Satoogol = B(10,100,6) ~ 10^^^9,803
Saptoogol = B(10,100,7) ~ 10^^^^101
Ashtoogol = B(10,100,8) ~ 10^^^^9,704
Navoogol = B(10,100,9) ~ 10^^^^^101
Dashoogol = B(10,100,10) ~ 10^^^^^9,607
Ekadashoogol = B(10,100,11) ~ 10^^^^^^101
Dvadashoogol = B(10,100,12) ~ 10^^^^^^9,512
Trayodashoogol = B(10,100,13) ~ 10{7}101
Chaturdashoogol = B(10,100,14) ~ 10{7}9,414
Panchadashoogol = B(10,100,15) ~ 10{8}101
Shodashoogol = B(10,100,16) ~ 10{8}9,324
Saptadashoogol = B(10,100,17) ~ 10{9}101
Ashtadashoogol = B(10,100,18) ~ 10{9}9,232
Navadashoogol = B(10,100,19) ~ 10{10}101
Vimshatoogol = B(10,100,20) ~ 10{10}9,139
Trimshatoogol = B(10,100,30) ~ 10{15}8,700
Chatvarimshatoogol = B(10,100,40) ~ 10{20}8,300
Panchashatoogol = B(10,100,50) ~ 10{25}7,910
Shastoogol = B(10,100,60) ~ 10{30}7,560
Saptatoogol = B(10,100,70) ~ 10{35}7,200
Ashitoogol = B(10,100,80) ~ 10{40}6,850
Navatoogol = B(10,100,90) ~ 10{45}6,600
Shatoogol = B(10,100,100) ~ 10{50}6,200
Sahasroogol = B(10,100,1000) ~ 10{500}1,000
Ayutoogol = B(10,100,10000) ~ 10{5000}1,000
Cintalum = B(10,100,1,2) = B(10,10,100) ~ 10{50}10
Expambaur = B(10,100,2,2)
Multpambaur = B(10,100,3,2)
Powpambaur = B(10,100,4,2)
Cintatrum = B(10,100,1,3) = B(10,10,100,2)
Explontaur = B(10,100,2,3)
Cinterum = B(10,100,1,4) = B(10,10,100,3)
Detonaur = B(10,100,2,4)
Cimpetum = B(10,100,1,5) = B(10,10,100,4)
Diamum = B(10,100,1,1,2) = B(10,10,1,100) = B(10,10,100,99)
Diabum = B(10,100,1,1,3) = B(10,10,1,100,2) = B(10,10,100,99,2)
Triamum = B(10,100,1,1,1,2) = B(10,10,100,99,99)
Tetramum = B(10,100,1,1,1,1,2) = B(10,10,100,99,99,99)
Linenbach = B(10,100{2}2)
Linentrach = B(10,100{2}3)
Linengold = B(10,100{2}1,2)
Deunen = B(10,100{2}1{2}2)
Xapplen = B(10,100{3}2)
Colossen = B(10,100{4}2)
Gongen = B(10,100{1,2}2) = B(10,10{100}2)
Gingen = B(10,100{1,3}2) = B(10,10{100,2}2)
Bongen = B(10,100{1,1,2}2) = B(10,10{1,100}2) = B(10,10{100,99}2)
Guplexen = B(10,100{1{2}2}2)
Gudexen = B(10,100{1{1,2}2}2)