A Flanking Pattern in a Sum-of-Divisors Congruence

TL;DR

This paper investigates a specific congruence involving the sum-of-divisors function and Euler's totient, revealing a unique 'flanking' pattern centered around the number 14 and providing a new characterization of the solutions.

Contribution

It uncovers a novel 'flanking' pattern in the solutions to a sum-of-divisors congruence and introduces a new characterization of the integers satisfying this condition.

Findings

14 appears in both neighboring sets whenever certain n are in the middle set

14 is the only nontrivial case of the flanking property

A new characterization of n in the solution sets is derived

Abstract

We consider composite $n$ satisfying the congruence $$n \cdot \sigma_k(n) \equiv 2 \pmod{\phi(n)},$$ 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 $S_{k}$.