Arithmetic properties of biregular overpartitions

Abstract In this paper, we investigate the arithmetic properties of $$\overline{B}_{\ell _{1}, \ell _{2}}(n)$$ B ¯ ℓ 1 , ℓ 2 ( n ) , the number of overpartitions where no part is divisible by $$\ell _{1}$$ ℓ 1 or $$\ell _{2}$$ ℓ 2 , with $$\ell _{1}$$ ℓ 1 and $$\ell _{2}$$ ℓ 2 being relatively prime. Specifically, we establish congruences modulo 3 and powers of 2 for the pairs $$(\ell _{1}, \ell _{2})\in \{(4,3), (4,9), (8,3), (8,9)\}$$ ( ℓ 1 , ℓ 2 ) ∈ { ( 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.