Dense admissible sequences

Abstract

A sequence of integers in an interval of length x 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 x is denoted by ϱ ∗ ( x ) \varrho ^{*}(x) . Hensley and Richards showed that ϱ ∗ ( x ) > π ( x ) \varrho ^{*}(x)>\pi (x) for large enough x x . We increase the known bounds on the set of x x satisfying ϱ ∗ ( x ) ≤ π ( x ) \varrho ^{*}(x)\le \pi (x) and find smaller values of x x such that ϱ ∗ ( x ) > π ( x ) \varrho ^{*}(x)>\pi (x) . We also find values of x x satisfying ϱ ∗ ( x ) > 2 π ( x / 2 ) \varrho ^{*}(x)>2\pi (x/2) . This shows the incompatibility of the conjecture π ( x + y ) − π ( y ) ≤ 2 π ( x / 2 ) \pi (x+y)-\pi (y)\le 2\pi (x/2) with the prime k k -tuples conjecture.