On the Number of Subsets of [1,m ] Relatively Prime to N and Asymptotic Estimates
نویسنده
چکیده
A set A of positive integers is relatively prime to n if gcd(A∪{n}) = 1. Given positive integers l ≤ m ≤ n, let Φ([l,m], n) denote the number of nonempty subsets of {l, l +1, . . . ,m} which are relatively prime to n and let Φk([l,m], n) denote the number of such subsets of cardinality k. In this paper we give formulas for these functions for the case l = 1. Intermediate consequences include identities for the number of subsets of {1, 2, . . . , n} with elements in both {1, 2, . . . ,m} and {m,m + 1, . . . , n} which are relatively prime to n and the number of such subsets having cardinality k. Some of our proofs use the Möbius inversion formula extended to functions of several variables.
منابع مشابه
Asymptotic estimates for phi functions for subsets of {m
Let f (m, n) denote the number of relatively prime subsets of {m+ 1, m + 2,. .. , n}, and let Φ(m, n) denote the number of subsets A of {m + 1, m + 2,. .. , n} such that gcd(A) is relatively prime to n. Let f k (m, n) and Φ k (m, n) be the analogous counting functions restricted to sets of cardinality k. Simple explicit formulas and asymptotic estimates are obtained for these four functions. A ...
متن کاملAffine Invariants , Relatively Prime Sets , and a Phi Function for Subsets of { 1 , 2 , . . . , N }
A nonempty subset A of {1, 2, . . . , n} is relatively prime if gcd(A) = 1. Let f(n) and fk(n) denote, respectively, the number of relatively prime subsets and the number of relatively prime subsets of cardinality k of {1, 2, . . . , n}. Let Φ(n) and Φk(n) denote, respectively, the number of nonempty subsets and the number of subsets of cardinality k of {1, 2, . . . , n} such that gcd(A) is rel...
متن کاملAsymptotic Estimates for Phi Functions for Subsets of { M + 1 , M + 2 , . . . , N }
Let f(m,n) denote the number of relatively prime subsets of {m + 1,m + 2, . . . , n}, and let Φ(m,n) denote the number of subsets A of {m+1,m+2, . . . , n} such that gcd(A) is relatively prime to n. Let fk(m,n) and Φk(m,n) be the analogous counting functions restricted to sets of cardinality k. Simple explicit formulas and asymptotic estimates are obtained for these four functions. A nonempty s...
متن کاملOn Relatively Prime Sets Counting Functions
This work is motivated by Nathanson’s recent paper on relatively prime sets and a phi function for subsets of {1, 2, 3, . . . , n}. We establish enumeration formulas for the number of relatively prime subsets and the number of relatively prime subsets of cardinality k of {1, 2, 3, . . . , n} under various constraints. Further, we show how this work links up with the study of multicompositions. ...
متن کاملTHE NUMBER OF RELATIVELY PRIME SUBSETS AND PHI FUNCTIONS FOR { m , m + 1 , . . . , n } Mohamed
The work in this paper is inspired and motivated by some work of Nathanson. We count the number of relatively prime subsets and the number of relatively prime subsets having some fixed cardinality that are in {m,m+1, . . . , n}. We also count the number of nonempty subsets of {m,m+1, . . . , n} whose gcd is relatively prime to n and the number of nonempty subsets {m,m+1, . . . , n} having some ...
متن کامل