  In this paper, we consider the set $\textit{NR}(G)$ of natural numbers which
are not in the numerical semigroup generated by a compound sequence $G$. We
generalize a result of Tuenter which completely characterizes $\textit{NR}(G)$.
We use this result to compute Sylvester sums, and we give a direct application
to the computation of weights of higher-order Weierstrass points on some
families of complex algebraic curves.
