Primarial functions are based on two types of functions : R (pronounced "er") and Я (pronounced "ya"), created by Licorneuhh[1] [2].


Definition

"Let Pi be the i-th prime number in order of natural numbers.


R(n) = Я(n) = Pn# for all n ∈ N*, Pn# being the primorial


R*(n) = Pn↑Pn-1↑Pn-2↑.↑.↑.↑3↑2 = PnPn-1..32 for all \(n \in \mathbb{N}\)*


Я*(n) = 2↑3↑.↑.↑.↑Pn-2↑Pn-1↑Pn = 23..Pn-1Pn for all \(n \in \mathbb{N}\)*


Now assume k be the amount of stars for each R*k(n) and Я*k(n).


R*k(n) = PnkPn-1kPn-2k.↑k.↑k.↑k3↑k2 for all \(n \in \mathbb{N}\)*


Я*k(n) = 2↑k3↑k.↑k.↑k.↑kPn-2kPn-1kPn for all \(n \in \mathbb{N}\)*,

using the Knuth's up-arrow notation."[1]

An update has been done, replacing all the "for all \(n \in \mathbb{N}\)*" by "for all \(n \in \mathbb{N}\)\{0}", surely for more clarity and exactitude.[2]

Numbers

Numbers has been coined with that functions [3] :

Heads-Unprimol = Tails-Unprimol = R(1) = Я(1) = 2


Heads-Biprimol = Tails-Biprimol = R(2) = Я(2) = 6


Heads-Triprimol = Tails-Triprimol = R(3) = Я(3) = 30


Heads-Tetprimol = Tails-Tetprimol = R(4) = Я(4) = 210


Heads-Pentprimol = Tails-Pentprimol = R(5) = Я(5) = 2310


Heads-Hexprimol = Tails-Hexprimol = R(6) = Я(6) = 30030


Heads-Un-1-primol = Tails-Un-1-primol = R*(1) = Я*(1) = 2


Heads-Bi-1-primol = R*(2) = 9


Heads-Tri-1-primol = R*(3) = 1 953 125


Heads-Tet-1-primol = R*(4) = 71953125 ≈ 1.286x101650582


Heads-Pent-1-primol = R*(5) = 1171953125 ≈ 1010106.2176


Heads-Hex-1-primol = R*(6) = 13R*(5) ≈ 101010106.2176

Heads-Bi-2-primol = R**(2) = 3↑↑2 = 27

Heads-Tri-2-primol = R**(3) = 5↑↑3↑↑2 = 5↑↑27

Heads-Tet-2-primol = R**(4) = 7↑↑5↑↑3↑↑2 = 7↑↑5↑↑27 = 7↑↑R**(3)

Heads-Pent-2-primol = R**(5) = 11↑↑7↑↑5↑↑27 = 11↑↑R**(4)

Heads-Hex-2-primol = R**(6) = 13↑↑11↑↑7↑↑5↑↑27 = 13↑↑R**(5)

Heads-Bi-3-primol = R***(2) = 3↑↑↑2 = 7 625 597 484 987

Heads-Tri-3-primol = R***(3) = 5↑↑↑7 625 597 484 987

Heads-Tet-3-primol = R***(4) = 7↑↑↑5↑↑↑7 625 597 484 987 = 7↑↑↑R***(3)

Heads-Pent-3-primol = R***(5) = 11↑↑↑7↑↑↑5↑↑↑7 625 597 484 987 = 11↑↑↑R***(4)

Heads-Hex-3-primol = R***(6) = 13↑↑↑11↑↑↑7↑↑↑5↑↑↑7 625 597 484 987 = 13↑↑↑R***(5)


Tails-Bi-1-primol = Я*(2) = 8


Tails-Tri-1-primol = Я*(3) = 2243 = 14134776518227074636666380005943348126619871175004951664972849610340958208 ≈ 1.413x1073


Tails-Tet-1-primol = Я*(4) = 2378125 ≈ 101037274.5766


Tails-Pent-1-primol = Я*(5) = 2351977326743 ≈ 101010109.14


Tails-Hex-1-primol = Я*(6) = 235734522712143931 ≈ 101010101013.465

Tails-Bi-2-primol = Я**(2) = 2↑↑3 = 16

Tails-Tri-2-primol = Я**(3) = 2↑↑3↑↑5 = 2↑↑(3↑3↑7625597484987)

Tails-Tet-2-primol = Я**(4) = 2↑↑3↑↑5↑↑7

Tails-Pent-2-primol = Я**(5) = 2↑↑3↑↑5↑↑7↑↑11

Tails-Hex-2-primol = Я**(6) = 2↑↑3↑↑5↑↑7↑↑11↑↑13

Tails-Bi-3-primol = Я***(2) = 2↑↑↑3 = 65536

Tails-Tri-3-primol = Я***(3) = 2↑↑↑3↑↑↑5

Tails-Tet-3-primol = Я***(4) = 2↑↑↑3↑↑↑5↑↑↑7

Tails-Pent-3-primol = Я***(5) = 2↑↑↑3↑↑↑5↑↑↑7↑↑↑11

Tails-Hex-3-primol = Я***(6) = 2↑↑↑3↑↑↑5↑↑↑7↑↑↑11↑↑↑13

In general:

Heads- means R function, Tails- means Я function,

A numerical infix represents the argument of the function,

A number represents the number of stars,

And -primol goes at the end.

References

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