This page is just for random functions that don't make the cut for the main page.
Short for Graphs of Square States.
GOSS(n) = Amount of possible n x n grids using n+1 colors.
Examples: GOSS(1) = 2, since it is a single square, which can be one of two colors.
GOSS(2) = 81, as there are 81 possible grids if the the grid is 2x2 and there are 3 colors.
Here are numbers I've made:
GOSS(100) = Centigoss
GOSS(200) = Ducentigoss
GOSS(300) = Tricentigoss
GOSS(1,000,000) = Megagoss
GOSS(GOSS(1,000)) = Kilogosogoss (pronounced kill-oh-gohs-oh-gahs)
MGOSS(n) is to apply GOSS to n, n times. Example: MGOSS(3) = GOSS(GOSS(GOSS(3))) = GOSS(GOSS(262,144))
[1]GOSS(n) = GOSS(n)
[2]GOSS(n) = MGOSS(n)
[3]GOSS(n) = MGOSSn(n)
[x]GOSS(n) = [x-1]GOSSn(n)
EXTENSION:
[x, 2]GOSS(n) = [[[...[[x]GOSS(n)]GOSS(n)...]GOSS(n)]GOSS(n)]GOSS(n) with x nestings
[x, a]GOSS(n) = [[[...[[x, a-1]GOSS(n), a-1]GOSS(n)..., a-1]GOSS(n), a-1]GOSS(n), a-1]GOSS(n) with x nestings
N-FOLD(n) takes n copies of n, and concatenates them together.
Examples: N-FOLD(2) = 22
N-FOLD(3) = 333
N-FOLD(4) = 4,444
N-FOLD(5) = 55,555
N-FOLD(10) = 10,101,010,101,010,101,010
PDF(n) = n^the amount of digits in n.
PDF(3) = 3^1 = 3
PDF(10) = 10^2 = 100
PDF(500) = 500^3 = 12,50,00,000
PDF(1,000,000) = 1,000,000^7 = 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000
PDF(PDF(1,000,000)) = 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000^57
Mega CLAN function. MCLAN(n) the largest number possible with MCLAN(n-1) letters applied to MCLAN(n-1) in CLAN. MCLAN(1) = the largest number possible with 10 letters applied to 10 in CLAN.
DEN(n) The largest finite number possibly definable using n*10 letters, creating only real words from the 2025 Oxford English Dictionary.
ESoCC(n) = The largest (finite) number possible in SoCC using at most n symbols. Pairs of parenthesis count as 1 symbol, ..(c).. counts as 2, commas don't count. [0]ESoCC(n) = ESoCC(n)
None of the numbers below are confirmed or proven
ESoCC(1) = | = 1
ESoCC(2) = |~ = || = 2
ESoCC(3) = ||~ = ||| = 3
ESoCC(4) = |||| = 4
ESoCC(5) = ||||| = 5
ESoCC(6) = |||||| = 6
ESoCC(7) = ||||||| = 7
ESoCC(8) = |||~~~|| = |||~~||| = 9
ESoCC(9) = |||~~~~|| = |||~~~||| = |||~~|||~~||| = |||~~||||||||| = 27
ESoCC(10) = |||~~~~~|| = |||~~~~||| = |||~~~|||~~~||| = |||~~~||||||||||||||||||||||||||| = |||~~~27 = 3^27 ≈ 7.6*10^12
ESoCC(11) = |||~~~~~~|| = |||~~~~~||| = |||~~~~|||~~~~|||
ESoCC(19) = H||||||,WHa:a~~..(a)..~~a = ||||||~~~~~~||||||
ESoCC(20) = H|||||||,WHa:a~~..(a)..~~a = |||||||~~~~~~~|||||||
ESoCC(28) = H||||,WHa:a~~..(H(a~°|))..~~a,WH|:| = ||||~~..(|||~~~~|||)..~~||||
[x]ESoCC(n) = [x-1]ESoCC([x-1]ESoCC(...[x-1]ESoCC([x-1]ESoCC(n))...)) with n [x-1]ESoCC()'s
[ω]ESoCC(n) = [n]ESoCC(n)
[ω+1]ESoCC(n) = [ω]ESoCC([ω]ESoCC(...[ω]ESoCC([ω]ESoCC(n))...)) with n nestings
[ω+a]ESoCC(n) = [ω+(a-1)]ESoCC([ω+(a-1)]ESoCC(...[ω+(a-1)]ESoCC([ω+(a-1)]ESoCC(n))...)) with n nestings
[ω+ω]ESoCC(n) = [ω2]ESoCC(n) = [ω+(n-1)]ESoCC([ω+(n-1)]ESoCC(...[ω+(n-1)]ESoCC([ω+(n-1)]ESoCC(n))...)) with n nestings
[ω2]ESoCC(3) = [ω+2]ESoCC([ω+2]ESoCC([ω+2]ESoCC(3)))
[ω3]ESoCC(n) = [ω+ω+ω]ESoCC(n) = [ω+ω+(n-1)]ESoCC([ω+ω+(n-1)]ESoCC(...[ω+ω+(n-1)]ESoCC([ω+ω+(n-1)]ESoCC(n))...)) with n nestings
[ω3]ESoCC(3) = [ω+ω+2]ESoCC([ω+ω+2]ESoCC([ω+ω+2]ESoCC(3)))
[ω^2]ESoCC(n) = [ω*ω]ESoCC(n) = [ωn]ESoCC(n)
[ω^2]ESoCC(3) = [ω3]ESoCC(3) = [ω+ω+2]ESoCC([ω+ω+2]ESoCC([ω+ω+2]ESoCC(3)))
[ω^b]ESoCC(n) = [ω*ω*ω...ω*ω*ω]ESoCC(n) with b ω's
[ω^ω]ESoCC(n) = [ω^^2]ESoCC(n) = [ω*ω*ω...ω*ω*ω]ESoCC(n) with n ω's
and so on, ω being able to be modified by any operator or function.
PSF(n) = π * (10^n)
PSF(0) = π
PSF(1) = 31.41592...
PSF(2) = 314.15926...
PSF(3) = 3141.59265...
PSF(PSF(1)) = 8.1859648e+31
PSF(PSF(PSF(1))) = π * 10^(8.1859648e+31)
PRF(n) = PSF(n) rounded to the nearest whole number.
# is any or empty array. Examples are provided under the more complicated parts of the definition.
Ulthomegra(n, #) = ULTM(n, #)
ULTM(0) = Biggams
ULTM(n) = 3[⎔]ω[⎔]ω...ω[⎔]ω[⎔]ω with ULTM(n-1) ω's in THOM
ULTM(n, 1, 1, 1...(k 1's)...1, 1, 1) = ULTM(n, n...(k-1 n's)...n, n, n, ULTM(n, n...(k-1 n's)...n, n, n, ...ULTM(ULTM(n, n, n...(k n's)...n, n, n))...) with ULTM(n) nestings
ULTM(3, 1) = ULTM(ULTM(ULTM(3)))
ULTM(5, 1, 1, 1) = ULTM(5, 5, ULTM(5, 5, ULTM(5, 5, ULTM(5, 5, ULTM(5, 5, 5)))))
ULTM(n, a, #) = ULTM(ULTM(...ULTM(ULTM(n, a-1, #), a-1, #)..., a-1, #), a-1, #) with n nestings
ULTM(3, 2) = ULTM(ULTM(ULTM(3, 1), 1), 1)
ULTM(n, 1, 1, 1...(k 1's)...1, 1, 1, a, #) = ULTM(n, n, n, n...(k n's)...n, n, ULTM(n, n, n, n...(k n's)...n, n, ......ULTM(n, n, n, n...(k n's)...n, n, ULTM(n, n, n, n...(k+1 n's)...n, n, n, a-1, #), a-1, #)......, a-1, #), a-1, #) with n nestings. What would have been the k+1th n in each nesting is replaced with the nest.
ULTM(3, 1, 2) = ULTM(3, ULTM(3, ULTM(3, 3, 1), 1), 1)
ULTM(4, 1, 1, 1, 3) = ULTM(4, 4, 4, ULTM(4, 4, 4, ULTM(4, 4, 4, ULTM(4, 4, 4, 4, 2), 2), 2), 2)
ULTM([n]) = ULTM(n, n, n...n, n, n) with n n's
BULTM(0) = Biggams
BULTM(n) = BULTM(n-1)[⎔]ω[⎔]ω...ω[⎔]ω[⎔]ω with BULTM(n-1) ω's in THOM
Everything else is the same as ULTM.
n⇶ka = n⇶⇶...(k)...⇶⇶a
n⇶10 = n{n}n
n⇶k0 = n⇶k-1n⇶k-1n...n⇶k-1n⇶k-1n with n n's
n⇶k-1n⇶k-1n...n⇶k-1n⇶k-1n is worked through in a similar way to arrow notation, right to left.
3⇶3⇶3⇶3 is the same as 3⇶(3⇶(3⇶3)), 4⇶⇶3⇶⇶5⇶⇶2 is the same as 4⇶⇶(3⇶⇶(5⇶⇶2)).
n⇶ka = ((...(n⇶ka-1)⇶ka-1...)⇶ka-1)⇶ka-1 with n nestings
n⇶[0] = n⇶nn
n⇶[a] = ((...(n⇶[a-1])⇶[a-1]...)⇶[a-1])⇶[a-1] with n nestings
# is any or empty array
0⤳0 = 2
n⤳ka = n⤳⤳...(k)...⤳⤳a
n⤳10 = 2^n
n⤳k0 = n⤳k-1n⤳k-1n...n⤳k-1n⤳k-1n with n n's
n⤳ka# = ((...(n⤳ka-1#)⤳ka-1#...)⤳ka-1#)⤳ka-1# with n nestings
n⤳k0⤳k0⤳k0...(g 0's)...0⤳k0⤳k0 = n⤳kn⤳kn...(g-1 n's)...n⤳kn⤳k(n⤳kn⤳kn⤳...(g-1 n's)...n⤳kn⤳k(...n⤳kn⤳kn...(g-1 n's)...n⤳kn⤳k(n⤳kn⤳kn...(g n's)...n⤳kn⤳kn)...)) with n nestings
n⤳k0⤳k0⤳k0...(g 1's)...0⤳k0⤳ka = n⤳kn⤳kn...(g n's)...n⤳kn⤳k(n⤳kn⤳kn⤳...(g n's)...n⤳kn⤳k(...n⤳kn⤳kn...(g n's)...n⤳kn⤳k(n⤳kn⤳kn...(g+1 n's)...n⤳kn⤳kn⤳ka-1)⤳ka-1...)⤳ka-1)⤳ka-1 with n nestings
n⤳[0] = n⤳nn⤳nn...n⤳nn⤳nn with n n's
n⤳[a] = ((...(n⤳[a-1])⤳[a-1]...)⤳[a-1])⤳[a-1] with n nestings
a⇃ = (((a-1)⇃ + (a-1)⇃) * (a-1)⇃)^(a-1)⇃
where 1⇃ = 1 and 2⇃ = ((1 + 1) * 1)^1 = 2
3⇃ = ((2 + 2) * 2)^2 = 64
4⇃ = ((64 + 64) * 64)^64 ~ 2.86389e+250
Hyparporial
a↾ = (...(((((a-1)↾ + (a-↿)↾) * (a-1)↾)^(a-1)↾)^^(a-1)↾)^^^(a-1)↾...)^^^...(a)...^^^(a-1)↾
where 1↾ = 1 and 2↾ = 2
3↾ = ((((2 + 2) * 2)^2)^^2)^^^2 = (64^^2)^^^2
n[!] = n!
n[[...[[!]]...]] = (n-1)[[...[[!]]...]] * 3n where 1[[...[[!]]...]] = 1
n[!!!...(k !'s)...!!!] = n[[..[[!!!...(k-1 !'s)...!!!]]...]] with n sets of []
n[[...[[!!!...(k !'s)...!!!]]...]] with x sets of [] = ((...(n[[...[[!!!...(k !'s)...!!!]]...]])[[...[[!!!...(k !'s)...!!!]]...]]...)[[...[[!!!...(k !'s)...!!!]]...]])[[...[[!!!...(k !'s)...!!!]]...]] with n nestings, with x-1 sets of [] in each
GF(1) = 1
GF(2) = 2
GF(n) = n!GF(n-1) in nested factorial notation.
Where # is any or empty array:
G(n) = f[f[...f[f[ω](ω)](ω)...](ω)](n) in the fast-growing hierarchy with n nestings, where [...] means subscript.
G(1) = f[ω](1) = 2
G(2) = f[f[ω](ω)](2) = f[f[ω](2)](2) = f[8](2)
...
G(n, 0) = G(G(...G(G(n))...)) with n nestings
G(n, a, #) = G(G(...G(G(n, a-1, #), a-1, #)..., a-1, #), a-1, #)) with n nestings
G(n, 0, 0, 0...(a 0's)...0, 0, 0) = G(n, n, n...(a-1 n's)...n, n, G(n, n, n...(a-1 n's)...n, n, ...G(n, n, n...(a-1 n's)...n, n, G(n, n, n...(a n's)...n, n, n))...)) with n nestings
G(n, 0, 0, 0...(a 0's)...0, 0, 0, b, #) = G(n, n, n...(a n's)...n, n, G(n, n, n...(a n's)...n, n, ...G(n, n, n...(a+1 n's)...n, n, n, b-1, #)..., b-1, #), b-1, #) with n nestings
G([n]) = G(n, n, n...n, n, n) with n n's
GCS(n, x, c, k, j, h) is a sequence. Step 1 is each integer from n down to 1 listen in descending order. Each step after, an operator is applied to each integer, and 1 is subtracted from the rightmost entry. Only apply the operators to entries that have not been subtracted from. Apply operator directly to numbers, and write it out. Apply direct operators to written out operators. After subtracting from 1, rather than being zero, the entry is removed. Entries are also added to the front of the chain every step depending on what k and j are. They are what the leftmost entry would be in the next step, except keeping the same operators as the rest of the chain. In GCS(n, x, c, k, j, h), the n is the main number, or the starting number. The x is the number used by the operator, which c. If c = 1, then the operator is addition. If it is 2, then multiplication, and so on. j is the operator used for the number of entries added, and k is the operand. For example if k is 2 and j is 3, the number of entries would square each time. h says for how far to add new entries (to ensure termination), as in after the full decay of h entries, stop adding more. GCS(n) = GCS(n, n, n, n, n, n, n). GCS([n]) = GCS(GCS(...GCS(GCS(n))...)) with n nestings. GCS([[...[[n]]...]]) with x sets of brackets = GCS([[...GCS([[......GCS([[...GCS([[...[[n]]...]])...]])...]]...)...]]) with n nestings with x-1 sets of brackets in each. GCS({n}) = GCS([[...[[n]]...]]) with n brackets.
Examples:
GCS(3, 2, 2, 1, 1, 2):
3, 2, 1
12*2, 6*2, 4*2, 1
48*4*2, 24*4*2, 12*4*2, 8*4*2
192*8*4*2, 96*8*4*2, 48*8*4*2, 24*8*4*2, 1022
...
GCS(3, 1, 1, 2, 2, 1)
3, 2, 1
7+1, 6+1, 5+1, 4+1, 3+1, 2
14+2+1, 13+2+1, 12+2+1, 11+2+1, 10+2+1, 9+2+1, 8+2+1, 7+2+1, 6+2+1, 5+2+1, 4+2+1, 1
...
nF = The product of all Fibonacci numbers up to the nth Fibonacci number.
Examples:
1F = 1
2F = 1
3F = 2
4F = 6
5F = 30
6F = 240
7F = 3120
...
nF = nF1G
nFxG = All Fibonacci numbers (up to the nth) together, exluding the 1's, using the xth hyper-operator.
Examples:
1FxT = 1
2FxT = 1
3F2T = 2
4F3T = 2↑3 = 8
5F3T = 2↑3↑5 = 2↑243
6F4T = 2↑↑3↑↑5↑↑8
nFωT = All Fibonacci numbers (up to the nth) together, excluding the 1's, using the kth hyper-operator where k is the current Fibonaxxi number.
Examples:
1FωT = 1
2FωT = 1
3FωT = 2
4FωT = 2↑3 (exponentiation is the 3rd hyper-operator)
5FωT = 2↑(3↑↑5) (tetration is the 4th hyperoperator)
6FωT = 2↑(3↑↑(5↑↑↑8)) (pentation is the 5th hyperoperator)
nF[ω]G = All Fibonacci numbers (up to the nth) together, excluding the 1's, using the kth hyper-operator where k is the current Fibonacci number, but in reverse.
Examples:
1F[ω]T = 1
2F[ω]T = 1
3F[ω]T = 2
4F[ω]T = 2↑↑3 = 2↑(2↑2) = 16 (tetration is the 4th hyperoperator)
5F[ω]T = 2↑↑↑(3↑↑5) (pentation is the 5th hyperoperator)
6F[ω]T = 2↑↑↑↑(3↑↑↑(5↑↑8)) (hexatation is the 6th hyperoperator)
nPF = All Fibonacci numbers that are also prime numbers up to the nth Fibonacci prime multiplied together
nEPF = All Fibonacci numbers that are also prime numbers up to the nth Fibonacci prime stacked in a power tower
Examples:
3PF = 2 * 3 * 5 = 30
4PF = 2 * 3 * 5 * 13 = 30 * 13 = 390
5PF = 2 * 3 * 5 * 13 * 89 = 390 * 89 = 34,710
6PF = 34,710 * 233 = 8,087,430
7PF = 8,087,430 * 1597 = 12,915,625,710
2EPF = 2^3 = 8
3EPF = 2^(3^5) > 10^73
4EPF = 2^(3^(5^13)) = 2^(3^1,220,703,125)
nPFX = All Fibonacci numbers up to the nth Fibonacci that are also prime numbers that can also be expressed as some (k!)-1 multiplied together.
nEPFX = All Fibonacci numbers up to the nth Fibonacci that are also prime numbers that can also be expressed as some (k!)-1 Stacked in a power tower.
The only known number in either of these sequences is 5. It is Fibonacci, prime, and can be expressed as (3!)-1
The other numbers (if any (there probably are)) are way too big to be calculated with our current systems.
nPFEX = All Fibonacci numbers up to the nth Fibonacci that are also prime numbers that can also be expressed as some (k!)-1 where k is itself some Fibonacci number multiplied together.
nEPFEX = All Fibonacci numbers up to the nth Fibonacci that are also prime numbers that can also be expressed as some (k!)-1 where k is itself some Fibonacci number stacked in a power tower.
Again, the only known number in these sequences is 5. It is Fibonacci, prime, and can be expressed as (3!)-1, since 3 is a Fibonacci number.
nMPF = All Fibonacci numbers up to the nth Fibonacci that are also prime numbers that can also be expressed as some (k!)-1 where k is itself some Fibonacci number that is also prime, multiplied together.
Again, the only known number in this sequence is 5. It is Fibonacci, prime, and can be expressed as (3!)-1, since 3 is a Fibonacci number and prime.
GP(n) = All Fibonacci numbers that are also prime numbers up to the nth Fibonacci prime (that are also a member in a pair of twin primes) that can also be expressed as (k!)-1 (where k is the smallest possible double-Fibonacci number (Fibonacci(Fibonacci(k)) where Fibonacci(k) = the kth term in the Fibonacci sequence) to fit these descriptions that hasn't been used yet for any previous F(n) numbers) that also contains at least k amount of k's within it's digits, multiplied together.
Where "f" is some function, "n" and "x" are variables.
RIF(f, 1, n) = f↑nn(n) using Miter's Iteration Notation
RIF(!, 1, n) = n!↑nn
RIF(↑, 1, n) = n↑^n(n)n
RIF(f, x, n) = f↑RIF(f, x-1, n)n(n)
RIF(!, 1, n) = n!↑RIF(!, x-1, n)n
RIF(↑, x, n) = n↑^RIF(↑, x-1, n)(n)n
RIF(f) = RIF(f, f(10), f(10))
RIF↑k(f) = RIF(f, RIF↑k-1(f), RIF↑k-1(f))
RIF(RIF, x, n) is not valid. In order to use the RIF function within itself, it must be done like so:
RIF(RIF, f, x, n) where f is the function within the nested RIF, where f is NOT RIF itself
Where RIF(RIF, f, 1, n) = RIF↑[RIF(f)]n(f)
RIF(RIF, f, x, n) = RIF↑[RIF(RIF, f, x-1, n)](f)
Examples:
RIF(RIF, GOSS, 2, 2) = RIF↑RIF(RIF, GOSS, 1, 2)2(GOSS)
RIF(RIF, GOSS, 1, 2) = RIF↑RIF(GOSS)2(GOSS)
RIF(RIF, GOSS, 10, 3) = RIF↑RIF(RIF, GOSS, 9, 3)2(GOSS)
nΛ = nΛ0 = Normal bouncing factorial
From this point on, "1"s are completely ignored and removed.
nΛ1 = 2^(3^(...n-1^(n^(n-1^(n-2^...3^(2^(3^...^n-2^(n-1^(n-2^(...n-3^(.........)...))))))))))
nΛa = The same as nΛ1 but with a ^'s between each number
nΛ[1] = nΛn
nΛ[a] = ((...(nΛ[a-1])Λ[a-1]...)Λ[a-1])Λ[a-1] with n nestings
Examples:
3Λ1 = 2^(3^2)
4Λ1 = 2^(3^(4^(3^(2^(3^2)))))
5Λ1 = 2^(3^(4^(5^(4^(3^(2^(3^(4^(3^(2^(3^2)))))))))))
3Λ2 = 2^^(3^^2)
2^^(3^3) = 2^^27
4Λ2 = 2^^(3^^(4^^(3^^(2^^(3^^2)))))
3Λ[1] = 3Λ3
3Λ[2] = ((3Λ[1])Λ[1])Λ[1]
"n" = Each Fibonacci number up to the nth number to the power of the nth prime number, multiplied together.
"2" = 1^3 * 1^3 = 1
"3" = 1^5 * 1^5 * 2^5 = 32
"4" = 1^7 * 1^7 * 2^7 * 3^7 = 279,936
"5" = 2^11 * 3^11 * 5^11 ~ 1.77147E16
"6" = 2^13 * 3^13 * 5^13 * 8^13 ~ 8.76488338E30
"7" = 2^17 * 3^17 * 5^17 * 8^17 * 13^13 ~ 8.80754132E54
"8" = 2^19 * 3^19 * 5^19 * 8^19 * 13^19 * 21^19 ~ 3.24417211E91
""3"" = "32"
= 2^131 * 3^131 * 5^131...1,346,269^131 * 2,178,309^131
"[n]" = ""...""n""..."" with n pairs of " "
"[[...[[n]]...]]" with k sets of [ ] = "[[...[["[[...[[......"[[...[["[[...[[n]]...]]"]]...]]"......]]...]]"]]...]]" with n nestings and k-1 sets of [ ] in each
"{n}" = "[[...[[n]]...]]" with n sets of [ ]
"[3]" = """3""" = ""32""
""[3]"" = """"3""""
"[[3]]" = "["["[3]"]"]"
"{3}" = "[[[3]]]" = "[["[["[[3]]"]]"]]"
"{4}" = "[[[[4]]]]" = "[[["[[["[[["[[[4]]]"]]]"]]]"]]]"
"{2}" = 1
"{"3"}" = "{32}"
= "[[...[[32]]...]]" with 32 sets of [ ]
= "[[...[["[[...[[......"[[...[["[[...[[32]]...]]"]]...]]"......]]...]]"]]...]]" with 32 nestings and 31 sets of [ ] in each
"{"{3}"}" =
R[a, 2](n) = R[a, 1](n) = R[a, 0](n) = R[a](n)
R[0](n) = n↑↑2 = n^n
R[0, b](n) for b > 2 = R[n, b-1](n)
R[a+1](n) for a > 0 = R[a]↑↑R[a](n)(n)
R[0](1) = 1^1 = 1
R[0](2) = 2^2 = 4
R[0](3) = 3^3 = 27
R[0](4) = 4^4 = 256
R[0](5) = 5^5 = 3125
R[1](2) = R[0]↑↑R[0](2)(2) = R[0]↑↑4(2) = R[0]↑(R[0]↑(R[0]↑(R[0]↑4(2))(2))(2))(2)
= R[0]↑(R[0]↑(R[0]↑(R[0](R[0](R[0](R[0](2)))))(2))(2))(2)
= R[0]↑(R[0]↑(R[0]↑(R[0](R[0](R[0](4))))(2))(2))(2)
= R[0]↑(R[0]↑(R[0]↑(R[0](R[0](256)))(2))(2))(2)
= R[0]↑(R[0]↑(R[0]↑(R[0](256^256))(2))(2))(2)
R[1](3) = R[0]↑↑R[0](3)(3) = R[0]↑↑27(3)
R[0, 3](3) = R[3, 2](3) = R[2, 2](3) = R[1, 2]↑↑R[1, 2](3)(3)
R[a+1, b](n) for a > 0 = R[a, b]↑↑...(b)...↑↑R[a, b](n)(n)
R[#](n) = R[n](n)
R[a, #](n) = R[a, n](n)
R[#, #](n) = R[n, n](n)
R[#+1](n) = R[#]↑↑R[#](n)(n)
R[##](n) = R[R[#](#), R[#](#)](n)
R_a[k](n) = R_a(R_a(...R_a(R_a(n))...)) with n nestings
R_0(n) = n+1
R_a(n) for a > 0 = R_a-1[R_a-1[...R_a-1[R_a-1[n](n)](n)...](n)](n) with n nestings
R_#(n) = R_n(n)
R_3(3) = R_2[R_2[...R_2[R_2[3](3)](3)...](3)](3) with R_2[3](3) nestings
R_2[3](3) = R_2(R_2(R_2(3)))
R_2(3) = R_1[R_1[R_1[3](3)](3)](3)
R_1[3](3) = R_1(R_1(R_1(3)))
R_1(3) = R_0[R_0[R_0[3](3)](3)](3)
R_0[3](3) = 6
R_0[R_0[3](3)](3) = R_0[6](3) = 9
R_0[R_0[R_0[3](3)](3)](3) = R_0[9](3) = 12
R_1(3) = 12
R_1(R_1(3)) = R_1(12)
R_1(12) = R_0[R_0[R_0[R_0[R_0[R_0[R_0[R_0[R_0[R_0[R_0[R_0[12](12)](12)](12)](12)](12)](12)](12)](12)](12)](12)](12)](12)
R_1(12) = R_0[R_0[R_0[R_0[R_0[R_0[R_0[R_0[R_0[R_0[R_0[24](12)](12)](12)](12)](12)](12)](12)](12)](12)](12)](12)
R_1(12) = R_0[R_0[R_0[R_0[R_0[R_0[R_0[R_0[R_0[48](12)](12)](12)](12)](12)](12)](12)](12)](12)
R_1(12) = R_0[R_0[R_0[R_0[R_0[R_0[R_0[R_0[60](12)](12)](12)](12)](12)](12)](12)](12)
R_1(12) = R_0[R_0[R_0[R_0[R_0[R_0[R_0[72](12)](12)](12)](12)](12)](12)](12)
R_1(12) = R_0[R_0[R_0[R_0[R_0[R_0[84](12)](12)](12)](12)](12)](12)
R_1(12) = R_0[R_0[R_0[R_0[R_0[96](12)](12)](12)](12)](12)
R_1(12) = R_0[R_0[R_0[R_0[108](12)](12)](12)](12)
R_1(12) = R_0[R_0[R_0[120](12)](12)](12)
R_1(12) = R_0[R_0[132](12)](12)
R_1(12) = R_0[144](12)
R_1(12) = 156
R_1(12) = 12*(12+1) = 12*13 = 156
:{a}: = a/10
:{a, b}: = :{a}:/b
:{a, b, c}: (:{a, b}:)^c
:{3, 8, 2}: = (:{3, 8}:)^2 = (:{3}:/8)^2 = ((3/10)/8)^2 = 0.00140625
:{3, 8, 3}: = 0.000052734375
:{a, #, b, c}: where # is any array, and there are > 3 entries in the array = :{a, #, b, c}: = :{a, #, (b+1)^((b+1)^a), c-1}: where :{a, #, b, 0}: = :{a, #, (b+1)^((b+1)^a)}:
:{3, 8, 1, 2}: = :{3, 8, 2^(2^3), 1}: = :{3, 8, 256, 1}: = :{3, 8, 257^(257^3), 0}:
:{2, 3, 4, 5, 6}: = :{2, 3, 4, 6^(6^2), 5}: = :{2, 3, 4, 6^36, 5}:
:{a#0}: = :{a, a, a...a, a, a}: with a a's
:{a#b}: for b>0 = :{:{...:{:{a#b-1}:#b-1}:...#b-1}:#b-1}: with a nestings
O(n) Takes each digit of n, and organizes them from greatest to smallest, right to left. If the number is already sorted, it squares the number.
O(3) = 3^2 = 9
O(O(3)) = O(9) = 81
O(O(O(3))) = O(81) = 6561
O↑4(3) = O(6561) = 6651
O(59) = 95
O(15992791) = 99975211
PTO(n) = The first n digits of pi (including the leading 3) multiplied together.
PETO(n) = A power tower of the first n digits of pi (including the leading 3) replacing all 1's with the previous digit.
PTO(n, 1) = PTO(n)
PETO(n, 1) = PETO(n)
PTO(n, a) = The first n digits of pi (including the leading 3) multiplied together, clustered in groups of a digits
PTO(10, 2) = 31*41*59*26*53*58*97*93*23*84 ~ 1.04456767E17
PTO(11, 2) = 31*41*59*26*53*58*97*93*23*84*6 = 6.26740599E17
PETO(n) = A power tower of the first n digits of pi (including the leading 3) replacing all lone 1's with the previous digit, clustered in groups of a digits
PETO(4, 2) = 31^41 ~ 1.3990378E61
PETO(5, 2) = 31^(41^5)
PTO(n, #) = The first n digits of pi (including the leading 3) multiplied together, clustered in incrementing amounts of digits.
PTO(10, #) = 3*14*159*2653*58979*32384 = 3.38385356E16
Concat(a, b) = a#b
Concat(a, b, c) = a##...##b with c #'s
MConcat(a, 1) = MConcat(a) = Concat(a, a, a)
MConcat(a, b) for b > 1 = MConcat(MConcat(...MConcat(MConcat(a, b-1), b-1)..., b-1), b-1) with a nestings
MConcat([a]) = MConcat(a, a)
a#b = The concatenation of a and b.
Concat(10, 11) = 10#11 = 1011
a##..##b with k #'s = a##...##a##...##a...a##...##a##...##a with b a's and k-1 #'s between each. Expressions are evaluated from right to left.
3##3 = 3#3#3 = 333
3###3 = 3##3##3 = 3##(333) = 333...333 with 333 3's = MConcat(3)
MConcat(4) = 4####4 = 4###(4###(4###4)) = 4###(4###(4##(4##(4##4)))) = 4###(4###(4##(4##(4444))))
RAN(n) is to take the result of randomly rolling an n-sided die.
RAD(n) = is to apply the RAN(n) operator (where 1 = addition, 2 = multiplication, 3 = exponentiation, 4 = tetration, and so on) between 2 n's
RAD(3) when a 2 is rolled = 3*3 = 9
RAD(4) when a 3 is rolled = 4^4 = 256
RAD(5) when a 5 is rolled = 5↑↑↑5
RAD(100) when a 50 is rolled = 100↑↑...↑↑100 with 48 ↑'s
RUD(n) = RAD↑n(n) (using Miter's Iteration Notation)
RED(n) = RUD↑↑n(n)
ROD(n) = RED↑↑↑n(n)
[0]R(n) = RAN(n), [1]R(n) = RAD(n), [2]R(n) = RUD(n), [3]R(n) = RED(n), [4]R(n) = ROD(n)
[a]R(n) for a > 0 = [a-1]R↑a-1n(n)
[#]R(n) = [n]R(n)
[#+1]R(n) = [#]R↑n(n)
Only for use in salad numbers constructed with many steps.
SIS(x) is to apply step 1 to x, then step 1 and step 2, then step 1, 2, and 3, then step 1, 2, 3, and 4, and so on until its 1, 2, 3...a-1 where a is the number of steps up to the current step.
SIS(x, a) = SIS↑↑...↑↑x(x) with a arrows, using Miter's Iteration Notation.
SimDarn(n) = The largest well defined integer possible, which is able to be defined with exactly n words from the oxford english dictionary, using whichever version is most recent at the time of use, such that the number's entire definition is defined (can't simply call upon functions created elsewhere, has to contain their definition.)
Darn(n) = The largest well defined integer possible using any n characters, such that the number's entire definition is defined (can't simply call upon functions created elsewhere, has to contain their definition.)
TYULG(n) = 1.1^(1.2^(...((n/10)+1)...))
When n is not an integer, round up to the nearest integer above.
TYULG(1) = 1.1
TYULG(TYULG(1)) = TYULG(2)
TYULG(2) = 1.1^1.2 ~ 1.12116936414
TYULG(3) = 1.1^(1.2^1.3) ~ 1.12840171289
TYULG(4) = 1.1^(1.2^(1.3^1.4)) ~ 1.1320297963
TYULG(5) = 1.1^(1.2^(1.3^(1.4^1.5))) ~ 1.13462909475
TYULG(6) = 1.1^(1.2^(1.3^(1.4^(1.5^1.6)))) ~ 1.13736002526
TYULG(7) = 1.1^(1.2^(1.3^(1.4^(1.5^(1.6^1.7))))) ~ 1.14214710482
TYULG(8) = 1.1^(1.2^(1.3^(1.4^(1.5^(1.6^(1.7^1.8)))))) ~ 1.16876187065
TYULG(9) = 1.1^(1.2^(1.3^(1.4^(1.5^(1.6^(1.7^(1.8^1.9))))))) ~ 1.1^(1.2^(1.3^(1.4^(1.5^(1.6^(1.7^3.05504604131)))))) ~ 1.1^(1.2^(1.3^(1.4^(1.5^(1.6^5.05862009604))))) ~ 1.1^(1.2^(1.3^(1.4^(1.5^10.7786767137)))) ~ 1.1^(1.2^(1.3^(1.4^79.0734619897))) ~
1.1^(1.2^(1.3^(1.4^79.0734619897))) ~ 1.1^(1.2^(1.3^358797720165))
[0]TG(n) = TYULG(n)
[a]TG(n) = [a-1]TG([a-1]TG([a-1]TG(...[a-1]TG([a-1]TG(n)+1)+1...)+1)+1)+1 with n nestings
#TG(n) = [[...[[n]TG(n)]TG(n)...]TG(n)]TG(n) with n nestings
TIF(n) creates a THOM array where each entry in the array is (a digit of n)+1, starting with the first digit on the left, and going right.
Examples:
TIF(1) = ◭2◮ = 3
TIF(2) = ◭3◮ = 4
TIF(TIF(1)) = TIF(3) = 5
TIF(10) = ◭2, 1◮ = 4
TIF(15) = ◭2, 6◮
TIF(100) = ◭2, 1, 1◮
TIF(167809) = ◭2, 7, 8, 9, 1, 10◮
Spiral(1) = 1+1
Spiral(n) is to apply the Spiral(n-1) operator between 2 Spiral(n-1)'s, where the first operator is addition.
Spiral(2) = 2*2
Spiral(3) = 4↑↑4 = 4↑4↑4↑4 = Tritet Jr.
To "Spiral n in" is to Spiral↑↑n(n)
To "Spiral n in in... in in" (with x in's) is to Spiral↑↑...↑↑n(n) with x+1 ↑'s
To "Spiral n next" is to Spiral↑^n(n)
TO "Spiral n between" is to RIF(Spiral, n, n)
To "Spiral n out" is to RIF(Spiral, RIF(Spiral, ...RIF(Spiral, n, n)..., ......), RIF(Spiral, ...RIF(Spiral, n, n)..., ...RIF(Spiral, n, n)...)) with n nestings.
These can also be said as "n Spiraled in/next/between/out"
[k]SPO(1) = k Spiraled Out, and [k]SPO(n) for n>1 as SPO(n-1) Spiraled Out
Word(n) = 26^n
f(n) for f() that normally has 2 inputs = f(n, n)
Ex.: Sentence(n) = Sentence(n, n)
Sentence(n, a) = Word↑a(n)
Paragraph(n, a) = Sentence↑a(n)
Page(n, a) = Paragraph↑a(n)
Chapter(n, a) = Page↑a(n)
Book(n, a) = Chapter↑a(n)
Series(n, a) = Book↑a(n)
Bookshelf(n, a) = Series↑a(n)
Library(n, a) = Bookshelf↑a(n)
District(n, a) = Library↑a(n)