A Class of Exponential Sequences with Shift-Invariant Discriminators
نویسندگان
چکیده
The discriminator of an integer sequence s = (s(i))i≥0, introduced by Arnold, Benkoski, and McCabe in 1985, is the function Ds(n) that sends n to the least integer m such that the numbers s(0), s(1), . . . , s(n−1) are pairwise incongruent modulo m. In this note we present a class of exponential sequences that have the special property that their discriminators are shift-invariant, i.e., that the discriminator of the sequence is the same even if the sequence is shifted by any positive constant. 1 Discriminators Let m be a positive integer. If S is a set of integers that are pairwise incongruent modulo m, we say that m discriminates S. Now let s = (s(i))i≥0 be a sequence of distinct integers. For all integers n ≥ 1, we define Ds(n) to be the least positive integer m that discriminates the set {s(0), s(1), . . . , s(n−1)}. The function Ds(n) is called the discriminator of the sequence s. The discriminator was first introduced by Arnold, Benkoski, and McCabe [1]. They derived the discriminator for the sequence 1, 4, 9, . . . of positive integer squares. More recently, discriminators of various sequences were studied by Schumer and Steinig [11], Barcau [2], Schumer [10], Bremser, Schumer, and Washington [3], Moree and Roskam [8], Moree [6], Moree and Mullen [7], Zieve [13], Sun [12], Moree and Zumalacárrequi [9], and Haque and Shallit [5]. In all of these cases, however, the discriminator is based on the first n terms of a sequence, for n ≥ 2. Therefore, the discriminator can depend crucially on the starting point of a
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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 ...
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عنوان ژورنال:
- CoRR
دوره abs/1702.00802 شماره
صفحات -
تاریخ انتشار 2017