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

Abstract

A nonempty subset A ¹1; 2; : : : ; nº is relatively prime if gcd.A/ D 1. Let f .n/ denote the number of relatively prime subsets of ¹1; 2; : : : ; nº. The sequence given by the values of f .n/ is sequence A085945 in Sloane's On-Line Encyclopedia of Integer Sequences. In this article we show that f .n/ is never a square if n 2. Moreover, we show that reducing the terms of this sequence modulo any prime l ¤ 3 leads to a sequence which is not periodic modulo l.

Evaluation of Dentinal Changes Following Application of Three Different Desensitizing Agents

“…However the better results achieved with Colgate Sensitive Pro-Relief in our study are supported by other researchers. Most of the studies demonstrated a good occluding effect of Pro-Argin products on open dentin tubules [16,23,24] which also explain the results of the clinical studies showing an instant relief of dentin hypersensitivity [25]. The two active ingredients, arginine and calcium, are found naturally in saliva and might have a role in natural occlusion of the dentinal tubules and formation of the protective layer on the dentin surface [2].…”

A Remark on Relatively Prime Sets

“…In Section 5, we show how our method can be used to obtain Shonhiwa's results [14], [15] in a simpler and shorter way. We conclude this paper by giving a possible research related to the work of Ayad and Kihel [4], Tang [16], and Toth [17].…”

“…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…”

“…There are some interesting problems that can be investigated further. For example, combinatorial identities given by Bachraoui and Salim in [8], divisor-type functions by Toth [17], arithmetic properties of the sequence f (n) = f ([1, n]), and Φ(n) = Φ([1, n]) by Ayad and Kihel [4] and Min Tang [16] can be developed more. The possible research questions are the following:…”

“…(Ayad and Kihel [4] proved that f (n) is never a square for n ≥ 2.) (ii) What are congruence relations for f (n), f k (n), Φ(n), and Φ k (n)?…”