Journal of Integer Sequences, Vol. 21 (2018), Article 18.7.2

On a Family of Sequences Related to Chebyshev Polynomials

Andrew N. W. Hone
School of Mathematics, Statistics & Actuarial Science
Sibson Building
University of Kent
Canterbury CT2 7FS

L. Edson Jeffery
11918 Vance Jackson Road
San Antonio, TX 78230

Robert G. Selcoe
16214 Madewood Street
Cypress, TX 77429


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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