On the classification of integers n that divide ϕ(n)+σ(n)

The expressions ϕ(n)+σ(n)−3n and ϕ(n)+σ(n)−4n are unusual among linear combinations of arithmetic functions in that they each vanish on a nonempty set of composite numbers. In 1966, Nicol proved that the set A:={n|(ϕ(n)+σ(n))/n∈N⩾3} contains 2a⋅3⋅(2a−2⋅7−1) if and only if 2a−2⋅7−1 is prime and conjectured that A contains no odd integers. A 2008 paper by Luca and Sandor completely classifies the elements of A that have three distinct prime factors and observes that Nicol's conjecture holds for numbers with fewer than six distinct prime factors. In this paper we let AK denote the set of n∈A with exactly K distinct prime factors and present a computer-implementable algorithm that decides whether Nicol's conjecture holds for a given AK. Using this algorithm, we verify Nicol's conjecture for A6 and completely classify the elements of A4. We prove that all but finitely many n∈A4 have the form 2a⋅3⋅p3⋅p4, and that all but finitely many n∈A5 are divisible by 6 and not 9. In addition, we prove that every AK is contained in a finite union of sequences that each have the form {p1a1i⋯pkaki⋅u⋅wi}i=1∞, where k⩾1, p1,…,pk are distinct primes, and each aji as well as the least prime factor of wi go to infinity as i does.