Dense admissible sequences

Abstract:

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 $\varrho ^{*}(x)$. Hensley and Richards showed that $\varrho ^{*}(x)>\pi (x)$ for large enough $x$. We increase the known bounds on the set of $x$ satisfying $\varrho ^{*}(x)\le \pi (x)$ and find smaller values of $x$ such that $\varrho ^{*}(x)>\pi (x)$. We also find values of $x$ satisfying $\varrho ^{*}(x)>2\pi (x/2)$. This shows the incompatibility of the conjecture $\pi (x+y)-\pi (y)\le 2\pi (x/2)$ with the prime $k$-tuples conjecture.