The asymptotic estimates of the number of subsets of {1,2,…,m} r-relatively prime to n

Abstract

In this paper, Let A be a nonempty subset of $\lbrace1,2,...,n\rbrace$ of positive integers. A is r-relatively prime if greatest $r^{th}$ power common divisor of elements of A is 1. In this case we write $ gcd_r(A)=1$. A is r-relatively prime to n if greatest $r^{th}$ power common divisor of elements of A and n is 1. It is denoted as $gcd_r\left( A\cup\left\lbrace n\right\rbrace \right)=1$. For positive integers $k,l,m,n$ we write $\left[l,m \right]=\lbrace l,l+1,...,m \rbrace$ and for $l\leq m \leq n$, let $\Phi^{(r)}{(\left[l,m \right], n)}$ be the number of subsets of $\left[ l,m \right] $ which are r-relatively prime to n. The number of sets in $\Phi^{(r)}{(\left[l,m \right], n)}$ of cardinality $k$ is $\Phi_k^{(r)}{(\left[l,m \right], n)}$. In the present work we consider the case for $l=1$. That is we obtain the formulae for the functions $\Phi^{(r)}{(\left[1,m \right], n)}$ and $\Phi_k^{(r)}{(\left[1,m \right], n)}$ . We also obtain the exact formulae for the functions $U^{(r)}(m,n)$ which denotes the number of subsets of $\left[1,n \right]$, having the elements in both the sets $\left[1,m \right] $ and $\left[m,n \right] $ which are r-relatively prime to n and the number of sets in $U^{(r)}(m,n)$ of cardinality $k$ is $U_k^{(r)}(m,n)$.