Ackermann function ($A(m,n)$)

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.

For the displayed Ackermann function convention, one possible inverse-growth scale is $\alpha (n)=\min \{m\ge 1:A(m,m)\ge n\}$ for positive integers $n$. It gives $\alpha (n)=1$ through $3$, $2$ from $4$ through $7$, and $3$ from $8$ through $61$. Several inverse Ackermann function conventions occur in algorithm analysis; changing convention can shift these small values without changing the very slow growth relevant to the disjoint-set data structure. This inverse-growth definition does not invert a two-variable function as an ordinary one-variable inverse function.