Fuga-

Fuga- is a prefix devised by Allstair Cockburn and his children, including the then six-year-old Kieran Cockburn for the sole purpose of continuing the gar- prefix invented by his son as mentioned in his A Fuga Really Big NumbersA Fuga Really Big Numbers blog. It is the successor of the fz- prefix. Fuga- is a portmanteau of fugue and gar-, and it means “that number raised to that number that number of times,” as Cockburn considers how a five-year-old who recently learned about powers might say “Googleplex raised to the googleplex raised to the googleplex raised to the… (until the voice wears out)” to beat others in naming the larger number.
The intended expression for fuga-n is \(\underbrace{n ↑ n ↑ n ↑ ... ↑ n}_n\).
Since exponentiation is not power associaive, fuga- is ambiguous, as noticed by Stephan Houben. Fuga- is now used on a number \(n\) to indicate left-associative repeated exponentiation \(((...((n^n)^n)^n ... ^n)\) with n amount of n's, which is equivalent to both \(n^{n^{n-1}}\) and \(n\downarrow \downarrow n\) in down-arrow notation. The more familiar right-associative repeated exponentiation is now called the megafuga- prefix.
The first few values of fuga-n are 1, 4, 19,683, 3.4028*1038, 7.1821*10436, 8.0191*106050, and 8.6958*1099424. It is an example of a function with growth rates slightly above double-exponential.
== Sources ==
== See also ==
ja:フガ
Category:Fuga- family
Category:Numbers by Kieran Cockburn
Category:Numbers named by kids
Category:Numbers discussed by Alistair Cockburn
 
Category:Prefixes
Category:Functions
Category:Doubly exponential growth rate
Category:Tetrational growth rate