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)
# 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