  The covering number of a group $G$, denoted by $\sigma(G)$, is the size of a
minimal collection of proper subgroups of $G$ whose union is $G$. We
investigate which integers are covering numbers of groups. We determine which
integers $129$ or smaller are covering numbers, and we determine precisely or
bound the covering number of every primitive monolithic group with a degree of
primitivity at most $129$ by introducing effective new computational
techniques. Furthermore, we prove that, if $\mathscr{F}_1$ is the family of
finite groups $G$ such that all proper quotients of $G$ are solvable, then
$\mathbb{N}-\{\sigma(G):G\in \mathscr{F}_1\}$ is infinite, which provides
further evidence that infinitely many integers are not covering numbers.
Finally, we prove that every integer of the form $(q^m-1)/(q-1)$, where
$m\neq3$ and $q$ is a prime power, is a covering number, generalizing a result
of Cohn.
