Relatively prime partitions with two and three parts

A set A of positive integers is relatively prime if gcd(A) = 1. A partition of n is relatively prime if its parts form a relatively prime set. The number of partitions of n into exactly k parts is denoted by p(n, k) and the number of relatively prime partitions into exactly k parts is denoted by pΨ(n,k). In this note we give explicit formulas for pΨ(n,2) and pΨ(n,3) in terms of the prime divisors of n.