Perfect Overpartitions and Factorization of Integers

In his classic text, \emph{Combinatory Analysis}, MacMahon defined a perfect partition of a positive integer $n$ as a partition whose parts contain exactly one partition of every positive integer not exceeding $n$. In this paper we apply the same definition to overpartitions which are integer partitions with the additional property that the final occurrence of each part may be overlined. It turns out that perfect overpartitions are enumerated by ordered factorization functions in which the occurrence of 2 as a factor determines the presence of an overlined part.