For a class of sequences, defined as unions of intervals \([p_n, q_n]\) with the assumptions \(q_{n - 1}/q_n \to \rho\), \(p_n/q_n \to \sigma\), \(0 <\rho < \sigma < 1\), the lower and upper asymptotic densities and the logarithmic density are calculated. It is shown that the primes in such a set have the same relative densities within the set of primes.