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
