  Suppose $X$ is a smooth projective connected curve defined over an
algebraically closed field of characteristic $p>0$ and $B \subset X$ is a
finite, possibly empty, set of points. Booher and Cais determined a lower bound
for the $a$-number of a $\mathbf{Z}/p \mathbf{Z}$-cover of $X$ with branch
locus $B$. For odd primes $p$, in most cases it is not known if this lower
bound is realized. In this note, when $X$ is ordinary, we use formal patching
to reduce that question to a computational question about $a$-numbers of
$\mathbf{Z}/p\mathbf{Z}$-covers of the affine line. As an application, when
$p=3$ or $p=5$, for any ordinary curve $X$ and any choice of $B$, we prove that
the lower bound is realized for Artin-Schreier covers of $X$ with branch locus
$B$.
