R function

R function is a googological function introduced by Wikia user Hyp cos in 2013. R function is often considered to be an extremely powerful googological notation, and may be on par with Strong array notation if both were properly defined. However, R function in its original form is unformalised, and thus cannot have a strength at all. Subsequently, Hyp cos released Version 2 and Version 2.0.1 in an attempt to formalise the notation (Did he succeed?).

In 2020, an attempt to formalise the original R function was made by Wikia user 12AbBa (Zongshu Wu) in his web book. The definition is currently WIP.

At the core of R function is the operator "R", which is binary, left-associative, and operates on a positive integer (written to the left of "R") and an expression (written to the right of "R").

Terminology

12AbBa calls the positive integer before the "R" the base, and the positive integer after the "R" the recursive part. An rsymbol is a "symbol" that is allowed to appear in the recursive part. Note the rsymbols work differently than normal symbols, as "37" will be considered 2 symbols, but it is only 1 rsymbol.

Basic R function

Basic R Function
Type	Binary
Based on	Exponentiation
Growth rate	\(f_{\omega}(n)\)

In Basic R function, there is only one type of rsymbol: a positive integer. As said above, a positive integer is considered one rsymbol. The recursive part is also only allowed to contain one rsymbol. Therefore, for any \(n\):

\(nR0=10^n\)

\(nR(k+1)=nRkRk\dots Rk\) w/\(n\) \("Rk"\)'s, if \(k\ge 0\)

Examples

\(100R0=10^{100}=\) googol

\(100R1=100\underbrace{R0R0\dots R0}_{\text{100 R0's}}=\underbrace{10^{10^{⋰^{10^{100}}}}}_{\text{100 10's}}=\) grangol

\(100R2=100\underbrace{R1R1\dots R1}_{\text{100 R1's}}\approx\) greagol

\(100R3=100\underbrace{R2R2\dots R2}_{\text{100 R2's}}\approx\) gigangol