On 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. 1. Background Our paper is motivated by a recent paper of Nathanson [8] who defined a nonempty subset A of {1, 2, . . . , n} to be relatively prime if gcd(A) = 1. He defined f(n) to be the number of relatively prime subsets of {1, 2, . . . , n} and, for k ≥ 1, fk(n) to be the number of relatively prime subsets of {1, 2, . . . , n} of cardinality k. Further, he defined Φ(n) to be the number of nonempty subsets A of the set {1, 2, . . . , n} such that gcd(A) is relatively prime to n and, for integer k ≥ 1, Φk(n) to be the number of subsets A of the set {1, 2, . . . , n} such that gcd(A) is relatively prime to n and card(A) = k. He obtained explicit formulas for these functions and deduced asymptotic estimates. These functions were subsequently generalized by El Bachraoui [5] to subsets A ∈ {m + 1,m + 2, . . . , n} where m is any nonnegative integer, and then by Ayad and Kihel [3] to subsets of the set {a, a + b, . . . , a + (n− 1)b} where a and b are any integers. El Bachraoui [4] defined for any given positive integers l ≤ m ≤ n, Φ([l,m], n) to be the number of nonempty subsets of {l, l + 1, . . . ,m} which are relatively prime to n and Φk(l,m], n) to be the number of such subsets of cardinality k. He found formulas for these functions when l = 1 [4]. INTEGERS: 10 (2010) 466 2. Introduction It turns out that some of Nathanson’s results are special cases of number theoretic functions investigated by Shonhiwa. In [10], Shonhiwa defined and investigated the following functions and established the following result. Theorem 1 Let S k (n) = ∑ 1≤a1,a2,...,ak≤n (a1,a2,...,ak,m)=1 1; ∀n ≥ k ≥ 1, m ≥ 1 (1) Gk(n) = ∑ 1≤a1,a2,...,ak≤n (a1,a2,...,ak)=1 1; ∀n ≥ k ≥ 1, (2) Lk (n) = ∑ 1≤a1≤a2≤···≤ak≤n (11,a2,...,ak,m)=1 1; ∀n ≥ k ≥ 1, m ≥ 1 (3) and T k (n) = ∑ 1≤a1<a2<···<ak≤n (a1,a2,...,ak,m)=1 1; ∀n ≥ k ≥ 1, m ≥ 1. (4) Then S k (n) = ∑ d|m μ(d) ⌊n d ⌋k , Lk (n) = ∑ d|m μ(d)Lk (⌊n d ⌋) = ∑ d|m μ(d) (⌊n d ⌋ + k − 1 k ) , and T k (n) = ∑ d|m μ(d)T 1 k (⌊n d ⌋) = ∑ d|m μ(d) (⌊n d ⌋ k ) . From above, it follows that Φk(n) = T k = ∑ d|m μ (n d )d k ) (5) INTEGERS: 10 (2010) 467 and Φ(n) = n ∑ k=1 T k (n) = ∑ d|m μ(d)2 n d , (6) as shown therein and as proved in [8]. 3. Main Results The result obtained concerning the function Gk(n) in [10] is incorrect and we provide the correction below. The corrected result makes use of the following theorem [1]. Theorem 2 (Generalized Möbius inversion formula) If α is completely multiplicative we have G(x) = ∑ n≤x α(n)F (x n ) ⇐⇒ F (x) = ∑ n≤x μ(n)α(n)G (x n ) . We may now prove our first result as follows. Theorem 3 We have Gk(n) = ∑ j≤n μ(j) ⌊ n j ⌋k .