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

Given the multiplicative group Zn⁎ of units in the ring Zn:=Z/nZ, let Zn⁎⁎ denote the set of exceptional units in Zn, i.e. units u∈Zn⁎ satisfying 1−u∈Zn⁎. A subset of a finite group G containing all generators of any (cyclic) subgroup of G is called an atom of G. Let An⁎ denote the set of all atoms of Zn⁎. By means of An⁎⁎:={A∈An⁎:A⊂Zn⁎⁎}, the set Zn⁎⁎ trivially decomposes into atoms, i.e. Zn⁎⁎=⋃A∈An⁎⁎ as a disjoint union. An explicit construction of that atom decomposition is easily obtained if n is a prime power.
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.