  By work of Bringmann, Ono, and Rhoades it is known that the generating
function of the $M_2$-rank of partitions without repeated odd parts is the
so-called holomorphic part of a certain harmonic Maass form. Here we improve
the standing of this function as a harmonic Maass form and show more can be
done with this function. In particular we show the related harmonic Maass form
transforms like the generating function for partitions without repeated odd
parts (which is a modular form). We then use these improvements to determine
formulas for the rank differences modulo $7$. Additionally we give identities
and formulas that allow one to determine formulas for the rank differences
modulo $c$, for any $c>2$.
