  The appearance of primes in a family of linear recurrence sequences labelled
by a positive integer $n$ is considered. 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. It is proved 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 $n$, it is conjectured 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.
