Overpartition pairs modulo powers of 2

An overpartition of nn is a non-increasing sequence of positive integers whose sum is nn in which the first occurrence of a number may be overlined. In this article, we investigate the arithmetic behavior of bk(n)bk(n) modulo powers of 22, where bk(n)bk(n) is the number of overpartition kk-tuples of nn. Using a combinatorial argument, we determine b2(n)b2(n) modulo 88. Employing the arithmetic of quadratic forms, we deduce that b2(n)b2(n) is almost always divisible by 2828. Finally, with the aid of the theory of modular forms, for a fixed positive integer jj, we show that b2k(n)b2k(n) is divisible by 2j2j for almost all nn.