Overpartition pairs modulo powers of 2

AbstractAn 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 bk(n) modulo powers of 2, where bk(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 b2k(n) is divisible by 2j for almost all n