A Flanking Pattern in a Sum-of-Divisors Congruence

We consider composite $n$ satisfying the congruence $$n\cdot {\sigma }_k(n)\equiv 2(\mathrm{m}\mathrm{o}\mathrm{d}\phi (n)),$$ and show a “flanking” structure: $14$ appears in both $S_{k-1}$ and $S_{k+1}$ whenever certain values of $n$ appear in $S_k$; and, moreover, $14$ is the only (nontrivial) case of this property. Along the way, we derive a new characterization of the $n$ that appear in the sets $S_k$.