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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 $$g_{\varphi(\omega, 0)}(n)$$