Observations on the Parity of the Total Number of Parts in Odd-part Partitions

In recent years, numerous functions which count the number of parts of various types of partitions have been studied. In this brief note, we consider the function pto(n) which counts the total number of parts in all odd–part partitions of n (or what Chen and Ji recently called the number of rooted partitions of n into odd parts). In particular, we prove a number of results on the parity of pto(n), including infinitely many Ramanujan–like congruences satisfied by the function.