A note on divisibility sequences

Abstract

A sequence of rational integers (u): u₀, u₁, u₂, . . .u_n, . . . is called a divisibility sequence if u_r divides u_s whenever r divides s, and any integer M dividing terms of (u) with positive suffix is called a divisor of (u). The suffix s is called a rank of apparition of M if u_s ≡ 0 (mod M), but u_r /≡ (mod M) if r is a proper divisor of s. It follows from a previous note of mine in this Bulletin (Ward [l]) that a necessary and sufficient condition that every divisor of (u) shall have only one rank of apparition is that (u) have the following property.