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) 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 involvinga(m,1)(+)(j) and the divisor function sigma(1)(n). At the end of his paper, Merca posed a number of congruences on a(m,1)(+)(n). In this paper, we confirm some conjectures due to Merca on the parity of a(m,1)(+)(n)by using some theta function identities. In particular, we establish infinite families congruences modulo 2 for a(17,1)(+)(n)