On a family of simple skew braces

Several constructions have been given for families of simple braces, but few examples are known of simple skew braces which are not braces. In this paper, we exhibit the first example of an infinite family of simple skew braces which are not braces and which do not arise from nonabelian simple groups. More precisely, we show that, for any primes $p$, $q$ such that $q$ divides $(p^p-1)\mathrm{/}(p-1)$, there are exactly two simple skew braces (up to isomorphism) of order $p^{p}q$.