  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.
