  A positive integer $n$ is a Cayley number if every vertex-transitive graph of
order $n$ is a Cayley graph. In 1983, Dragan Maru\v{s}i\v{c} posed the problem
of determining the Cayley numbers. In this paper we give an infinite set $S$ of
primes such that every finite product of distinct elements from $S$ is a Cayley
number. This answers a 1996 outstanding question of Brendan McKay and Cheryl
Praeger, which they "believe to be the key unresolved question" on Cayley
numbers.
  We also show that, for every finite product $n$ of distinct elements from
$S$, every transitive group of degree $n$ contains a semiregular element.
