Completely torsion-free, finite-rank, almost decomposable groups with torsion factor

This paper deals with almost completely decomposable finite rank groups $G$ that have rank $1$ summands of pairwise noncomparable types. It is well known that every such group has unique complete quasi-decomposition $A$ with respect to equality. We consider the number of almost completely decomposable groups $G$ with a given quasi\df decomposition $A$ for which $G/A$ is isomorphic to $\mathbb{Z}(p^m)$.