On the Characterization of $\tau_{(n)}$-Atoms

In 2011, Anderson and Frazier define the concept of $\tau_{(n)}$-factorization, where $\tau_{(n)}$ is a restriction of the modulo $n$ equivalence relation. These relations have been worked mostly for small values of $n$. However, it is sometimes difficult to extend findings to larger values of $n$. One of these problems is finding $\tau_{(n)}$-irreducible elements or $\tau_{(n)}$-atoms in order to characterize elements that have a $\tau_{(n)}$-factorization in $\tau_{(n)}$-atoms. The $\tau_{(n)}$-irreducible elements are well known for $n=0,1,2,3,4,5,6,8,10,12$. However, the problem of determining the $\tau_{(n)}$-atoms becomes much more difficult the larger $n$ is. In this work, we present an algorithm to construct families of $\tau_{(n)}$-atoms. It is shown that the algorithm terminates in finitely many steps when $n$ is the safe prime associated to a Sophie Germain prime.