2010/02/17 by Mohamed El Bachraoui, Bachraoui, Mohamed El
Mathematics · #11A25 #11B05 #11B75 #Advanced Mathematical Identities #Advanced Mathematical Theories #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11A25 #msc:11B05 #msc:11B75
paper · pdf · doi:10.48550/arxiv.1002.3254
arxiv created 2010/02/17 · openalex publication_date 2010/02/17 · arxiv updated 2010/02/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let n be a positive integer and let A be nonempty finite set of positive integers. We say that A is relatively prime if gcd(A) =1 and that A is relatively prime to n if gcd(A,n)=1. In this work we count the number of nonempty subsets of A which are relatively prime and the number of nonempty subsets of A which are relatively prime to n. Related formulas are also obtained for the number of such subsets having some fixed cardinality. This extends previous work for the cases where A is an interval or a set in arithmetic progression. Applications include: a) An exact formula is obtained for the number of elements of A which are co-prime to n; note that this number is ϕ(n) if A=[1,n]. b) Algebraic characterizations are found for a nonempty finite set of positive integers to have elements which are all pairwise co-prime and consequently a formula is given for the number of nonempty subsets of A whose elements are pairwise co-prime. c) We provide combinatorial formulas involving Mertens function.