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

G. Lalitha, G. Kamala

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 $U_k^{(r)}(m,n)$.

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Published: 2018-11-19

How to Cite this Article:

G. Lalitha, G. Kamala, The asymptotic estimates of the number of subsets of {1,2,…,m} r-relatively prime to n, Advances in Inequalities and Applications, Vol 2018 (2018), Article ID 16

Copyright © 2018 G. Lalitha, G. Kamala. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Advances in Inequalities and Applications

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