Overpartition pairs modulo powers of 2

An overpartition of n is a non-increasing sequence of positive integers whose sum is n in which the first occurrence of a number may be overlined. In this article, we investigate the arithmetic behavior of b_k(n) modulo powers of 2, where b_k(n) is the number of overpartition k-tuples of n. Using a combinatorial argument, we determine b₂ (n) modulo 8. Employing the arithmetic of quadratic forms, we deduce that b₂ (n) is almost always divisible by 2⁸. Finally, with the aid of the theory of modular forms, for a fixed positive integer j, we show that b_2k(n) is divisible by 2^j for almost all n.