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On a Family of Sequences Related to Chebyshev Polynomials
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Andrew N. W. Hone

School of Mathematics, Statistics & Actuarial Science

Sibson Building

University of Kent

Canterbury CT2 7FS

UK

L. Edson Jeffery

11918 Vance Jackson Road

San Antonio, TX 78230

USA

Robert G. Selcoe

16214 Madewood Street

Cypress, TX 77429

USA

**Abstract:**

We consider the appearance of primes in a family of linear recurrence sequences
labelled by a positive integer *n*. The terms of each sequence correspond to a particular
class of Lehmer numbers, or (viewing them as polynomials in *n*) dilated versions of the
so-called Chebyshev polynomials of the fourth kind, also known as airfoil polynomials.
We prove that when the value of *n* is given by a dilated Chebyshev polynomial of the
first kind evaluated at a suitable integer, either the sequence contains a single prime,
or no term is prime. For all other values of *n*, we conjecture that the sequence contains
infinitely many primes, whose distribution has analogous properties to the distribution
of Mersenne primes among the Mersenne numbers. Similar results are obtained for
the sequences associated with negative integers *n*, which correspond to Chebyshev
polynomials of the third kind, and to another family of Lehmer numbers.

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(Concerned with sequences
A000032
A000045
A001519
A001834
A001906
A002315
A002327
A002878
A005248
A005478
A008865
A030221
A033890
A045546
A057080
A057081
A088165
A113501
A117522
A269251
A269252
A269253
A269254
A285992
A294099
A298675
A298677
A298878
A299045
A299071
A299100
A299101
A299107
A299109.)

Received February 8 2018; revised versions received July 23 2018; July
31 2018; August 2 2018. Published in *Journal of Integer
Sequences*, August 23 2018.

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