Let \(\overline{p}_{k}(n)\) denote the number of overpartition k-tuples of n. In 2023, Saikia (Boletín de la Sociedad Matemática Mexicana 29:15, 2023) conjectured the following congruences: \(\begin{aligned} \overline{p}_{q}(8n+2)&\equiv 0 \pmod {4},\quad \overline{p}_{q}(8n+3) \equiv 0 \pmod {8},\quad \\ \overline{p}_{q}(8n+4)&\equiv 0 \pmod {2},\quad \overline{p}_{q}(8n+5) \equiv 0 \pmod {8}, \\ \overline{p}_{q}(8n+6)&\equiv 0 \pmod {8},\quad \overline{p}_{q}(8n+7)\equiv 0 \pmod {32}, \end{aligned}\) where \(n\ge 0\) and q is prime. Recently, Sellers (Boletín de la Sociedad Matemática Mexicana 30:2, 2024) 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 \(i,j\ge 1\) , \(n\ge 0\) , 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 \(\begin{aligned} \overline{OPT}_{2^i\cdot r}(8n+7)&\equiv 0 \pmod {2^{i+4}},\\ \overline{OPT}_{3^i\cdot 2^j\cdot k}(3n+2)&\equiv 0 \pmod {3^{i+1}\cdot 2^{j+2}},\\ \overline{OPT}_{3^i\cdot 2^j\cdot k}(3n+1)&\equiv 0 \pmod {3^{i}\cdot 2^{j+1}},\\ \overline{OPT}_{3^i\cdot \ell }(3n+2)&\equiv 0 \pmod {3^{i+1}\cdot 2},\\ \overline{OPT}_{3^i\cdot \ell }(3n+1)&\equiv 0 \pmod {3^{i}\cdot 2},\end{aligned}\) where the first congruence was posed as a conjecture by Sarma et al. (Arithmetic properties modulo powers of 2 for overpartition k-tuples with odd parts, 2023) and the latter four were conjectured by Das et al. (Arithmetic properties modulo powers of 2 and 3 for overpartition k-tuples with odd parts, 2024).
