Dense admissible sequences

A sequence of integers in an interval of length x is called admissible if for each prime there is a residue class modulo the prime which contains no elements of the sequence. The maximum number of elements in an admissible sequence in an interval of length x is denoted by %∗(x). Hensley and Richards showed that %∗(x) > π(x) for large enough x. We increase the known bounds on the set of x satisfying %∗(x) ≤ π(x) and find smaller values of x such that %∗(x) > π(x). We also find values of x satisfying %∗(x) > 2π(x/2). This shows the incompatibility of the conjecture π(x+y)−π(y) ≤ 2π(x/2) with the prime k-tuples conjecture.