Ackermann Function

This function is a computable function but not primitive recursive. It grows faster than an exponential function.

Definition

The recurrence relation the Ackermann Function is based off of is $A(m+1,n+1)=A(m,A(m+1,n))$. The value grows incredible fast. For example, the value of $A(4,2)$ is $2^{65536}-3$.

Inverse Ackermann Function

The inverse Ackermann Function (written as $\alpha(n)$) grows incredibly slowly, effectively making it constant. It's so slow that it is effectively $\leq5$ for all $n$, even when $n$ is as large as the number of atoms in the universe.