Journal of Integer Sequences, Vol. 23 (2020), Article 20.8.4

Characterizing Quasi-Friendly Divisors

C. A. Holdener
Grinnell College
Grinnell, IA 50112

J. A. Holdener
Department of Mathematics and Statistics
Kenyon College
Gambier, OH 43022


Abundancy ratios are rational numbers k satisfying σ(N)/N = k/m for some NZ≥1, where σ is the sum-of-divisors function. In this paper we examine abundancy ratios of the form σ(m)+1 , where gcd(m, σ(m) + 1) = 1. Defining D to be a quasi-friendly divisor of N if σ(N) = σ(D)+1 with D|N, our main results characterize all possible quasi-friendly divisors D having two or more distinct prime divisors and satisfying gcd(D, σ(D) + 1) = 1. In fact, we prove that no such quasi-friendly divisor can have more than two distinct prime divisors.

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(Concerned with sequences A240991.)

Received May 25 2020; revised versions received August 14 2020; August 22 2020. Published in Journal of Integer Sequences, September 2 2020.

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