Arithmetic density and congruences of -regular bipartitions

Let B_ (n) denote the number of -regular bipartitions of n. In this article, we prove that B_ (n) is always almost divisible by pᵢʲ if pᵢ^2aᵢ, where j is a fixed positive integer and =p₁^a₁p₂^a₂ pₘ^aₘ, where pᵢ are prime numbers 5. Further, we obtain an infinities families of congruences for B₃ (n) and B₅ (n) by using Hecke eigen form theory and a result of Newman Newmann1959. Furthermore, by applying Radu and Seller's approach, we obtain an algorithm from which we get several congruences for B₏ (n), where p is a prime number.