An integer n is said to be ternary if it is composed of three distinct odd primes. In this paper, we asymptotically count the number of ternary integers n ≤ x with the constituent primes satisfying various constraints. We apply our results to the study of the simplest class of (inverse) cyclotomic polynomials that can have coefficients that are greater than 1 in absolute value, namely to the nth (inverse) cyclotomic polynomials with ternary n. We show, for example, that the corrected Sister Beiter conjecture is true for a fraction ≥ 0.925 of ternary integers.