Arithmetic Properties For Overpartition Triples With Odd Parts

Let $ \overline{PT}_o(n)$ denote the number of overpartition triples of a positive integer $n$ into odd parts. In this paper, we prove some infinite families of congruences modulo small powers of 2  and 3 for $ \overline{PT}_o(n)$. For example, for any $\alpha\geq0$ and $1\leq r\leq p-1$,  we prove that$$\overline{PT}_o\Big(8\cdot p^{2\alpha+1}(pn+r)+3\cdot p^{2\alpha+2}\Big )\equiv 0 \pmod{16}.$$