Chebyshev polynomials, Lehmer numbers, and huge primes

We consider the appearance of prime numbers in a family of linear recurrence sequences,  labelled by a positive integer n. The terms of each sequence correspond to a particular type of number studied by Lehmer, or (viewed as polynomials in n) dilated versions of the so-called Chebyshev polynomials of the 4th kind, which appear in the analysis of fluid flow over an aerofoil. It turns out that when n is given by a dilated Chebyshev polynomial of the 1st 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. However, most of these (probable) primes are so huge that numerical  attempts to measure their distribution test the limits of standard computer algebra packages.