A combinatorial problem and numerical semigroups

Let $a=(a_1,\dots ,a_n)$ and $b=(b_1,\dots ,b_n)$ be two $n$-tuples of positive integers, let $X$ be a set of positive integers, and let $g$ be a positive integer. In this work we show an algorithmic process in order to compute all the sets $C$ of positive integers that fulfill the following conditions: 1) the cardinality of $C$ is equal to $g$; 2) if $x,y\in $\mathbb{N}$ \setminus {0}$and$x+y\in C$,then$C \cap {x,y} \neq \emptyset$;3)if$x \in C$and$\frac{x-b_i}{a_i} \in $\mathbb{N}$ \setminus {0}$forsome$i\in {1,\ldots,n}$,then$\frac{x-b_i}{a_i} \in C$;4)$X \cap C = \emptyset$.