On Relatively Prime Sets

Abstract

Functions counting the number of subsets of ¹1; 2; : : : ; nº having particular properties are defined by Nathanson. Here, generalizations in two directions are given.

A Remark on Relatively Prime Sets

“…In Section 3, we show that we need only to derive the formula for Φ k (X, n) as the others will follow as a consequence. This will cover the results in [2], [3], [5], [6], [12], and [13]. In Section 4, we extend the formulas obtained by Ayad and Kihel [2], [3], by El Bachraoui [7], [8], [9], and by El Bachraoui and Salim [10].…”

“…In this section, we give a simple proof of the formula for Φ (a,b) k (m, n) and show that the formulas for Φ (a,b) (m, n), f (a,b) k (m), and f (a,b) (m) can be obtained as a consequence. In the notation used in [2], [3]…”

“…Authors Formulas for f (X), f k (X), Φ(X, n), Φ k (X, n) Nathanson [12] X = [1, n] El Bachraoui [5] X = [m, n] Nathanson and Orosz [13] El Bachraoui [6] X = [1, m] X = {a, a + b, a + (n − 1)b}, Ayad and Kihel [2], [3] X = [ℓ, m], X = {a, a + b, a + (m − 1)b} El Bachraoui [7], [9] X = [1, m 1 ] ∪ [ℓ 2 , m 2 ] El Bachraoui and Salim [10] [14], [15] X = [1, n] with various contraints Ayad and Kihel [4], Different direction such as congruence El Bachraoui [8], [9] properties, divisor sum types, Tang [16], Toth [17] combinatorial identities…”

The Number of Relatively Prime Subsets of {1, 2, . . . , n}

“…Let f .n/ andˆ.n/ denote respectively the number of relatively prime subsets of ¹1; 2; : : : ; nº and the number of nonempty subsets A of ¹1; 2; : : : ; nº such that gcd.A/ is relatively prime to n. Exact formulas and asymptotic estimates are given by M. B. Nathanson in [5]. Generalizations may be found in [1], [2], [3], [4] and [6]. Let OEx denote the greatest integer less than or equal to x and .n/ the Möbius function.…”

“…In [1], a new function ‰.n; p/ The research of the second author is partially supported by NSERC. generalizingˆis defined such that ‰.n; p/ represents the number of primitive elements of F p n over F p , where p is any prime number.…”