Congruences modulo powers of 2 and 3 for overpartition $k$-tuples
Let $\overline{p}_{k}(n)$ denote the number of overpartition $k$-tuples of $n$. In 2023, Saikia \cite{saikia} conjectured the following congruences: \begin{align*} \overline{p}_{q}(8n+2)& \equiv 0 \pmod{4},\quad \overline{p}_{q}(8n+3)\equiv 0 \pmod{8},\quad \overline{p}_{q}(8n+4) \equiv 0 \pmod{2},\\ \overline{p}_{q}(8n+5)& \equiv 0 \pmod{8},\quad \overline{p}_{q}(8n+6) \equiv 0 \pmod{8},\quad \overline{p}_{q}(8n+7)\equiv 0 \pmod{32}, \end{align*} where $n\geq0$ and $q$ is prime. Recently, Sellers \cite{sellers2024elementary} showed that these congruences hold for all odd integers $q$ (not necessarily prime). In this paper, we show that the above congruences hold for all positive integers $q$ (not necessarily odd). We also prove the following congruences on $\overline{OPT}_k(n)$, the number of overpartition $k$-tuples with odd parts of $n$: For all $i,j\geq 1$, $n\geq 0$, $r$ not a multiple of 2, $k$ not a multiple of 2 or 3, and $\ell$ not a power of 2, nor a multiple of 2 or 3, we have \begin{align*} \overline{OPT}_{2^i\cdot r}(8n+7)& \equiv 0 \pmod{2^{i+4}}, \overline{OPT}_{3^i\cdot 2^j\cdot k}(3n+2)& \equiv 0 \pmod{3^{i+1}\cdot 2^{j+2}}, \overline{OPT}_{3^i\cdot 2^j\cdot k}(3n+1)& \equiv 0 \pmod{3^{i}\cdot 2^{j+1}},\\ \overline{OPT}_{3^i\cdot \ell}(3n+2)& \equiv 0 \pmod{3^{i+1}\cdot 2}, \overline{OPT}_{3^i\cdot \ell}(3n+1)& \equiv 0 \pmod{3^{i}\cdot 2},\end{align*} where the first congruence was posed as a conjecture by Sarma et al. \cite{saikiasarma} and the latter four were conjectured by Das et al. \cite{DSS}.