On Relatively Prime Subsets and Supersets

Abstract

A nonempty finite set of positive integers A is relatively prime if gcd.A/ D 1 and it is relatively prime to n if gcd.A [ ¹nº/ D 1. The number of nonempty subsets of A which are relatively prime to n isˆ.A; n/ and the number of such subsets of cardinality k isˆk.A; n/. Given positive integers l 1 , l 2 , m 2 , and n such that l 1 Ä l 2 Ä m 2 we givê .OE1; m 1 [ OEl 2 ; m 2 ; n/ along withˆk.OE1; m 1 [ OEl 2 ; m 2 ; n/. Given positive integers l; m, and n such that l Ä m we count for any subset A of ¹l; l C 1; : : : ; mº the number of its supersets in OEl; 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.

A Remark on Relatively Prime Sets

“…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 Section 5, we show how our method can be used to obtain Shonhiwa's results [14], [15] in a simpler and shorter way.…”

“…In this paper, we give simpler and shorter calculations which lead to the results extending those in the literature. To be precise, we cover the results of Nathanson [12], Nathanson and Orosz [13], El Bachraoui [5], [6], [7], [8], [9], El Bachraoui and Salim [10], Ayad and Kihel [2], [3], and Shonhiwa [14], [15]. We show how to apply our method to obtain all results mentioned above and their generalization.…”

“…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…”

Coprime partitions and Jordan totient functions

“…Formulas ( 17) and ( 18) seem encouraging in this direction. However, in [2] it is observed that the situation becomes very complicated for k ≥ 4 and no information is given for the general approach. Our paper aims to explain in which sense and why complications do arise.…”

“…Let J i denote the Jordan totient function of degree i ≥ 0. In [2], it is proved that p ′ 3 (n) = J 2 (n) 12 holds for n ≥ 4. It is also clear that p ′ 2 (n) = J 1 (n) 2 holds for n ≥ 3.…”