Representation of integers: a nonclassical point of view

In \cite{A.Boudaoud1}, the author asked the following question: Which $n\in \mathbb{N}$ unlimited can be represented as a sum $% n=s+w_{1}w_{2}$, where $s\in \mathbb{Z}$ is a limited integer and $\omega _{1}$, $\omega _{2}$ are two unlimited positive integers? In this survey article we partially answer this question, i.e., we present some families of unlimited positive integers which can be written as the sum of a limited integer and the product of at least two unlimited positive integers.