A Flanking Pattern in a Sum-of-Divisors Congruence
Scott Duke Kominers
TL;DR
The paper studies composite n solving n · σ_k(n) ≡ 2 (mod φ(n)) by refining the known structure of the exceptional sets S_k and uncovering a robust flanking pattern. It shows that for n = 2p with p an odd prime and p ≡ 3 (mod 4), 2p ∈ S_k ⇔ (p−1)/2 | (2^{k+1}+1), leading to precise periodic appearances of 2p in the S_k and, notably, that 14 is the unique nontrivial distance-1 flanker around certain 2p values (e.g., around 22). A complete analysis ties flanking to the multiplicative order ord_r(2) in (Z/rZ)× and yields a trichotomy depending on whether −1 lies in ⟨2⟩ and the parity of the least t0 with 2^{t0} ≡ −1 (mod r). Computational data up to p ≤ 10000 corroborates the flanking behavior (67 flanked 2p values), and conditional Hardy-Littlewood-type conjectures imply infinitely many such instances; the results point toward extensions to higher-distance flanking and tighter bounds on S_k.
Abstract
We consider composite $n$ satisfying the congruence $$n \cdot σ_k(n) \equiv 2 \pmod{φ(n)},$$ 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 $S_{k}$.