Proofs of some conjectural congruences of Merca on a function related to the sum of the parts congruent to 0 modulo m

Recently, Merca defined a function \(a_{m,1}^{+} (n)\) $a_{m,1}^{+}(n)$ which is related to the sum of the parts congruent to 0 modulo m in all the partitions of n. He also proved some identities involving \(a_{m,1}^{+} (j)\) $a_{m,1}^{+}(j)$ and the divisor function \(\sigma _1(n)\) ${\sigma }_1(n)$ . At the end of his paper, Merca posed a number of congruences on \(a_{m,1}^{+} (n)\) $a_{m,1}^{+}(n)$ . In this paper, we confirm some conjectures due to Merca on the parity of \(a_{m,1}^{+} (n)\) $a_{m,1}^{+}(n)$ by using some theta function identities. In particular, we establish infinite families congruences modulo 2 for \(a_{17,1}^{+} (n)\) $a_{17,1}^{+}(n)$ .