Amicable pairs with few distinct prime factors

Abstract

An amicable pair [Formula: see text] is a pair of distinct positive integers [Formula: see text] such that [Formula: see text], where [Formula: see text] is the sum of divisors function. In this note, we prove that if [Formula: see text] are amicable and [Formula: see text] is odd, then [Formula: see text], where [Formula: see text] is the number of distinct prime factors.