  An arithmetical structure on the complete graph $K_n$ with $n$ vertices is
given by a collection of $n$ positive integers with no common factor each of
which divides their sum. We show that, for all positive integers $c$ less than
a certain bound depending on $n$, there is an arithmetical structure on $K_n$
with largest value $c$. We also show that, if each prime factor of $c$ is
greater than $(n+1)^2/4$, there is no arithmetical structure on $K_n$ with
largest value $c$. We apply these results to study which prime numbers can
occur as the largest value of an arithmetical structure on $K_n$.
