Minimal order attaining function

Definition

The minimal order attaining function or moa function is the function $\mathrm{m}\mathrm{o}\mathrm{a}:\mathbb{N}\to \mathbb{N}$ defined by $\mathrm{m}\mathrm{o}\mathrm{a}(n)$ equal to the smallest number such that there are $n$ groups of that order up to isomorphism.

In terms of the group number function

We can define the minimal order attaining function in terms of the group number function, denoted \mathrm{gnu}, which outputs the number of groups of a given order up to isomorphism:

$\mathrm{m}\mathrm{o}\mathrm{a}(n)=\mathrm{m}\mathrm{i}\mathrm{n}\{m\in \mathbb{N}:\mathrm{g}\mathrm{n}\mathrm{u}(m)=n\}$.