In this paper, we investigate the arithmetic properties of \(\overline{B}_{\ell _{1}, \ell _{2}}(n)\) , the number of overpartitions where no part is divisible by \(\ell _{1}\) or \(\ell _{2}\) , with \(\ell _{1}\) and \(\ell _{2}\) being relatively prime. Specifically, we establish congruences modulo 3 and powers of 2 for the pairs \((\ell _{1}, \ell _{2})\in \{(4,3), (4,9), (8,3), (8,9)\}\) . We use generating functions, dissection formulas, and Smoot’s implementation of Radu’s Ramanujan–Kolberg algorithm to prove the main results.
