Discriminators and k-Regular Sequences

The discriminator of an integer sequence s = (s(i))i≥0, introduced by Arnold, Benkoski, and McCabe in 1985, is the map Ds(n) that sends n ≥ 1 to the least positive integer m such that the n numbers s(0), s(1), . . . , s(n − 1) are pairwise incongruent modulo m. In this note we consider the discriminators of a certain class of sequences, the k-regular sequences. We compute the discriminators of two such sequences, the socalled “evil” and “odious” numbers, and show they are 2-regular. We give an example of a k-regular sequence whose discriminator is not k-regular. Finally, we examine sequences that are their own discriminators, and count the number of length-n finite sequences with this property. 1 Discriminators Let s = (s(i))i≥0 be a sequence of distinct integers. For each n ≥ 1, if the n numbers s(0), s(1), . . . , s(n − 1) are pairwise incongruent modulo m, we say that m discriminates them. For n ≥ 1 we define Ds(n) to be the least positive integer m that discriminates the numbers s(0), s(1), . . . , s(n − 1); such an m always exists because of the distinctness requirement. Furthermore, we set Ds(0) = 0, but usually this will be of no consequence. The function (or sequence) Ds(n) is called the discriminator of the sequence s, and was introduced by Arnold, Benkoski, and McCabe [3]. They proved that the discriminator Dsq(n)