Multiplicative atom decomposition of sets of exceptional units in residue class rings

We characterise so-called tame integers, i.e. odd n>1 with prime factorisation n=∏piki, say, for which the atom decomposition of Zn⁎⁎ is obtained by multiplicative composition of the atom decompositions of the Zpiki⁎⁎. Moreover, it is shown that the set of tame integers has density zero.