On an Overpartition Analogue of $SOME(n)$

Recently, Andrews and Dastidar introduced the partition function $SOME(n)$, defined as the sum of all the odd parts in the partitions of $n$ minus the sum of all the even parts in the partitions of $n$. They derived its generating function and established some congruences satisfied by $SOME(n)$. In this paper, we introduce an overpartition analogue of $SOME(n)$, denoted by $\overline{SOME}(n)$, the sum of all the odd parts in the overpartitions of $n$ minus the sum of all the even parts in the overpartitions of $n$. We derive the generating function for $\overline{SOME}(n)$ and obtain congruences modulo $3, \ 5$ and powers of $2$. Our method is based on classical $q$-series identities and manipulations of infinite products and sums.