AckermanFunctions

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Ackerman Functions. Also known as Grzegorczyk functions. f_1, f_2,... from the naturals to the naturals. Defined as follows:

 f_1 (x) := 2x
 f_n (x) := f_(n-1)^(x) (1) == f_(n-1) of f_(n-1) of ... a total of x times ... of f_(n-1) of 1.

Some examples:

 f_2 (x) = 2^x
 f_3 (x) = 2^2^...^2 a total of x times
 f_4 (1) = 2
 f_4 (2) = 2^2 = 4
 f_4 (3) = 2^2^2^2 = tower of 4 2s = 2^16 = 65536
 f_4 (4) = 2^2^...total of 65536...^2 = tower of 65536 2s = large.

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CategoryMaths