Arithmetic Properties of Overpartition Triples

Let ${\bar{p}}_3(n)$ be the number of overpartition triples of $n$. By elementary series manipulations, we establish some congruences for ${\bar{p}}_3(n)$ modulo small powers of 2, such as $${\bar{p}}_3(16n+14)\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}32),{\bar{p}}_3(8n+7)\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}64)\mathrm{.}$$ We also find many arithmetic properties for ${\bar{p}}_3(n)$ modulo 7, 9 and 11, involving the following infinite families of Ramanujan-type congruences: for any integers $\alpha \ge 1$ and $n\ge 0$, we have ${{\overline{p}}{3}}\big({{3}^{2\alpha +1}}(3n+2)\big)\equiv 0$(mod$9\cdot 2^4$),$\overline{p}{3}(4^{\alpha-1}(56n+49)) \equiv 0$ (mod 7) and $${\bar{p}}_3(7^{2\alpha +1}(7n+3))\equiv {\bar{p}}_3(7^{2\alpha +1}(7n+5))\equiv {\bar{p}}_3(7^{2\alpha +1}(7n+6))\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}7)\mathrm{.}$$