The number of maximum primitive sets of integers

Abstract

AbstractA set of integers is primitive if it does not contain an element dividing another. Let f(n) denote the number of maximum-size primitive subsets of {1,…,2n}. We prove that the limit α = limn→∞f(n)1/n exists. Furthermore, we present an algorithm approximating α with (1 + ε) multiplicative error in N(ε) steps, showing in particular that α ≈ 1.318. Our algorithm can be adapted to estimate the number of all primitive sets in {1,…,n} as well.We address another related problem of Cameron and Erdős. They showed that the number of sets containing pairwise coprime integers in {1,…n} is between ${2^{\pi (n)}} \cdot {e^{(1/2 + o(1))\sqrt n }}$ and ${2^{\pi (n)}} \cdot {e^{(2 + o(1))\sqrt n }}$. We show that neither of these bounds is tight: there are in fact ${2^{\pi (n)}} \cdot {e^{(1 + o(1))\sqrt n }}$ such sets.