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:

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

Small values

$n$	$\mathrm{m}\mathrm{o}\mathrm{a}(n)$	Relevant "groups of order" page
1	1	groups of order 1
2	4	groups of order 4
3	75	groups of order 75
4	28	groups of order 28
5	8	groups of order 8
6	42	groups of order 42
7	375	groups of order 375
8	510	groups of order 510
9	308	groups of order 308
10	90	groups of order 90