The least primary factor of the multiplicative group

Let $S(n)$ denote the least primary factor in the primary decomposition of the multiplicative group $M_n=(\mathbb{Z}\mathrm{/}n\mathbb{Z}{)}^{\times }$. We give an asymptotic formula, with order of magnitude $x\mathrm{/}(\log x{)}^{1\mathrm{/}2}$, for the counting function of those integers $n$ for which $S(n)\mathrm{\ne }2$. We also give an asymptotic formula, for any prime power $q$, for the counting function of those integers $n$ for which $S(n)=q$. This group-theoretic problem can be reduced to problems of counting integers with restrictions on their prime factors, allowing it to be addressed by classical techniques of analytic number theory.