On Relatively Prime Subsets and Supersets

TL;DR

This paper derives formulas for counting subsets and supersets of integer sets that are relatively prime to a given number, extending previous results and providing new combinatorial insights into relatively prime subsets.

Contribution

It introduces new formulas for counting relatively prime subsets and supersets within specified integer ranges, including fixed cardinalities and intersections, advancing combinatorial number theory.

Findings

Formulas for (A, n) and _k(A, n) for specific integer ranges.

Counting supersets of a subset that are relatively prime to n.

Results include formulas for fixed cardinality supersets and intersections with fixed sets.

Abstract

A nonempty finite set of positive integers A is relatively prime if gcd(A) = 1 and it is relatively prime to n if gcd(A [ fng) = 1. The number of nonempty subsets of A which are relatively prime to n is \Phi(A, n) and the number of such subsets of cardinality k is \Phi_k(A, n). Given positive integers l1, l2, m2, and n such that l1 <= l2 <= m2 we give \Phi([1;m1][[l2;m2]; n) along with Phi_k([1;m1] [ [l2;m2]; n). Given positive integers l;m, and n such that l <= m we count for any subset A of {l,l+1,...,m} the number of its supersets in [l;m] which are relatively prime and we count the number of such supersets which are relatively prime to n. Formulas are also obtained for corresponding supersets having fixed cardinalities. Intermediate consequences include a formula for the number of relatively prime sets with a nonempty intersection with some fixed set of positive integers.