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The Ackermann Function bothers me, because it is a function of two variables, and we'd be interested in functions of one variable. Surely there must be a name for Ackermann(n,n) for whatever n. Knackered_Man(n) & Ackermann(n,n)?? Or, Gag(n) for short? And that +, *, ^ progression... does it generate new functions as it goes? Is that what tetration and quintation mean? Thanks for all this
  —Alistair Cockburn

The Knackeredman, or gag for short, is a function defined as \(\mathrm{gag}(n) = A(n,n)\), where \(A(x,y)\) stands for the Ackermann function. The name came from Alistair Cockburn (of fuga- glory) in one of his forum exchanges.[1] Due to Ackermann function's relation with Arrow notation , \(\text{Gag-x} = 2\uparrow^{x-2}(x+3)-3\). Since this function diagnalizes the Ackermann function, Knackeredman is not primitive recursive.

Unlike the gar-, fz-, and (mega)fuga-, Alistair Cockburn never mentioned gag (but did mention hyperoperations) in A fuga really big numbers[2]. He also never used gag as a prefix despite rumors of him doing so; the prefix form was coined by Sbiis Saibian, also known for coining booga-[3].

Approximations

Notation Approximation
Arrow notation \(2\uparrow^{x-2}(x+3)-3\)
Bowers' Exploding Array Function \(\{2,(n-2),(n+3)\}\)
Bird's array notation \(\{2,(n-2),(n+3)\}\)
Fast-growing hierarchy \(f_\omega(n)\)
Hardy hierarchy \(H_{\omega^\omega}(n)\)
Slow-growing hierarchy \(g_{\varphi(\omega, 0)}(n)\)

Sources[]

See also[]

Gar-: garone · gartwo · garthree · garfour · garfive · garsix · garseven · gareight · garnine · garten · garhundred · garmillion · gargoogol · gareceton · gartrialogue · gargoogolplex · gargiggol
Fz-: fzone · fztwo · fzthree · fzfour · fzfive · fzsix · fzseven · fzeight · fznine · fzten · fztwenty · fzthirty · fzhundred · fzthousand · fzmillion · fzgoogol · fzmilliplexion · fzgoogolplex · fzgargoogolplex · fzgargantugoogolplex · fzgiggol
Fuga-: fugaone · fugatwo · fugathree · fugafour · fugafive · fugasix · fugaseven · fugaeight · fuganine · fugaten · fugahundred · fugagoogol · fugagoogolplex · fugagargoogolplex · fugagargantugoogolplex · fugagiggol
Megafuga-: megafugaone · megafugatwo · megafugathree · megafugafour · megafugafive · megafugasix · megafugaseven · megafugaeight · megafuganine · megafugaten · megafugahundred · megafugagoogol · megafugagoogolplex · megafugagargoogolplex · megafugagargantugoogolplex · megafugagrangol
Extensions:
Booga-:: boogaone · boogatwo · boogathree · boogafour · boogafive · boogasix · boogaten · boogahundred · boogagoogol · boogagoogolplex
Gag-:: gagone · gagtwo · gagthree · gagfour · gagfive · gagsix · gagseven · gageight · gagnine · gagten · gaggoogol · gaggoogolplex

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