Congruences modulo powers of 2 and 3 for overpartition k-tuples

Let p₊ (n) denote the number of overpartition k-tuples of n. In 2023, Saikia saikia conjectured the following congruences: align* pₐ (8n+2) & 0 4, pₐ (8n+3) 0 8, pₐ (8n+4) 0 2, \\ pₐ (8n+5) & 0 8, pₐ (8n+6) 0 8, pₐ (8n+7) 0 32, align* where n0 and q is prime. Recently, Sellers 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 OPTₖ (n), the number of overpartition k-tuples with odd parts of n: For all i, j 1, n 0, r not a multiple of 2, k not a multiple of 2 or 3, and not a power of 2, nor a multiple of 2 or 3, we have align* OPT₂䂐 ₑ (8n+7) & 0 2^i+4, OPT₃䂐 ₂⋑ ₊ (3n+2) & 0 3^i+1 2^{j+2}, OPT₃䂐 ₂⋑ ₊ (3n+1) & 0 3^i 2^{j+1}, \\ OPT₃䂐 (3n+2) & 0 3^i+1 2, OPT₃䂐 (3n+1) & 0 3^i 2, align* where the first congruence was posed as a conjecture by Sarma et al. saikiasarma and the latter four were conjectured by Das et al. DSS.