Journal of Integer Sequences, Vol. 23 (2020), Article 20.11.3 |

Department of Mathematics and Applied Mathematics

University of Pretoria

Private Bag X20

Hatfield 0028

South Africa

Florian Luca

School of Maths

Wits University

1 Jan Smuts Avenue

Braamfontein 2000

Johannesburg

South Africa

and

Centro de Ciencias Matemáticas

UNAM, Morelia

Mexico

Lilit Martirosyan

Department of Mathematics and Statistics

University of North Carolina, Wilmington

601 South College Road

Wilmington, NC 28403-5970

USA

Maria Matthis

Department of Mathematics

Katharineum zu Lübeck

Königsstraße 27-31

23552 Lübeck

Germany

Pieter Moree

Max-Planck-Institut für Mathematik

Vivatsgasse 7

D-53111 Bonn

Germany

Max A. Stoumen

Department of Mathematics and Statistics

University of North Carolina, Wilmington

601 South College Road

Wilmington, NC 28403-5970

USA

Melvin Weiß

Department of Mathematics

Universität Bonn

Endenicher Allee 60

53115 Bonn

Germany

**Abstract:**

Given a sequence **w** =
(*w*_{n})_{n≥0} of distinct
positive integers *w*_{0}, *w*_{1},
*w*_{2},... and any positive integer *n*, we define
the discriminator function *D*_{w}(*n*) to be
the smallest positive integer *m* such that *w*_{0},
..., *w*_{n−1} are pairwise incongruent
modulo *m*. In this paper, we classify all binary recurrent
sequences *w* consisting of different integer terms such that
*D*_{w}(2^{e}) = 2^{e}
for every *e*
≥ 1. For all of these sequences it is expected that one can actually
give a fairly simple description of *D*_{w}(*n*)
for every *n* ≥ 1. For one infinite family of such sequences
this has already been done by Faye, Luca, and Moree, and for another by Ciolan
and Moree.

(Concerned with sequences A084222 A270151.)

Received March 10 2020; revised versions received September 28 2020; October 23 2020.
Published in *Journal of Integer Sequences*,
November 28 2020.

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