 Ackermann function

One convention for the Ackermann function on nonnegative integers isIts two-argument recursion defines a computable function; fixing the convention matters when defining an inverse Ackermann function.

The Ackermann function is a well-known example of a recursive function that is not primitive recursive. It serves as a benchmark for computing and illustrates the concept of deep recursion.

To get an intuition for it, see the sample computation at: en.wikipedia.org/w/index.php?title=Ackermann_function&oldid=1170238965#TRS,_based_on_2-ary_function where $S(n)=n+1$ in this context. From this, we immediately get the intuition that these functions are recursive somehow.