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 b 2 ( n ) modulo 8 . Employing the arithmetic of quadratic forms, we deduce that b 2 ( n ) is almost always divisible by 2 8 . Finally, with the aid of the theory of modular forms, for a fixed positive integer j , we show that b 2 k ( n ) is divisible by 2 j for almost all n .