Primarial Functions

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

Definition

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

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

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

Я*(n) = 2↑3↑.↑.↑.↑P_n-2↑P_n-1↑P_n = 2^3..P_n-1P_n 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) = P_n↑^kP_n-1↑^kP_n-2↑^k.↑^k.↑^k.↑^k3↑^k2 for all \(n \in \mathbb{N}\)*

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

using the Knuth's up-arrow notation."

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.

Numbers

Numbers has been coined with that functions  :

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) = 7^1953125 ≈ 1.286x10^1650582

Heads-Pent-1-primol = R*(5) = 11^71953125 ≈ 10^10106.2176

Heads-Hex-1-primol = R*(6) = 13^R*(5) ≈ 10^1010106.2176

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 ≈ 10^1037274.5766

Tails-Pent-1-primol = Я*(5) = 2^351977326743 ≈ 10^1010109.14

Tails-Hex-1-primol = Я*(6) = 2^{35734522712143931} ≈ 10^{1010101013.465}

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.