If A is a set of nonnegative integers containing 0, then there is a unique nonempty set B of nonnegative integers such that every positive integer can be written in the form a+b, where a\in A and b\in B, in an even number of ways. We compute the natural density of B for several specific sets A, including the Prouhet-Thue-Morse sequence, { 0} \cup { 2^n : n \geq 0}, and random sets, and we also study the distribution of densities of B for finite sets A. This problem is motivated by Euler's observation that if A is the set of n that have an odd number of partitions, then B is the set of pentagonal numbers { n(3n+1)/2 : n \in Z}. We also elaborate the connection between this problem and the theory of de Bruijn sequences and linear shift registers.
Categories: math.NT math.CO