Relatively Prime Sets, Divisor Sums, and Partial Sums

2013/06/20 by Prapanpong Pongsriiam, Pongsriiam, Prapanpong
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.1306.4891

submitted

arxiv created 2013/06/20 · openalex publication_date 2013/06/20 · arxiv updated 2013/06/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a nonempty finite set A of positive integers, let gcd(A) denote the greatest common divisor of the elements of A. Let f(n) and Φ(n) denote, respectively, the number of subsets A of \1, 2, …, n\ such that gcd(A) = 1 and the number of subsets A of \1, 2, …, n\ such that gcd(A∪\n\) =1. Let D(n) be the divisor sum of f(n). In this article, we obtain partial sums of f(n), Φ(n) and D(n). We also obtain a combinatorial interpretation and a congruence property of D(n). We give open questions concerning Φ(n) and D(n) at the end of this article.

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