Year: 2013
Type: preprint
Abstract: For a nonempty finite set $A$ of positive integers, let $\gcd\left(A\right)$ denote the greatest common divisor of the elements of $A$. Let $f\left(n\right)$ and $Φ\left(n\right)$ denote, respectively, the number of subsets $A$ of $\left\{1, 2, \ldots, n\right\}$ such that $\gcd\left(A\right) = 1$ and the number of subsets $A$ of $\left\{1, 2, \ldots, n\right\}$ such that $\gcd\left(A\cup\left\{n\right\}\right) =1$. Let $D\left(n\right)$ be the divisor sum of $f\left(n\right)$. In this article, we obtain partial sums of $f\left(n\right)$, $Φ\left(n\right)$ and $D\left(n\right)$. We also obtain a combinatorial interpretation and a congruence property of $D\left(n\right)$. We give open questions concerning $Φ\left(n\right)$ and $D\left(n\right)$ at the end of this article.
Source: arXiv (Cornell University)
Author: Pongsriiam, Prapanpong
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W1892073095