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\begin{center}
\vskip 1cm{\LARGE\bf 
The Number of Relatively Prime Subsets of a \\
\vskip .1in
Finite Union of Sets of Consecutive Integers
}
\vskip 1cm
\large
Mohamed Ayad\\
Laboratoire de Math\'ematiques Pures et Appliqu\'ees\\
Universit\'e du Littoral\\
F-62228 Calais \\
France\\
\href{mailto:Ayad@lmpa.univ-littoral.fr}{\tt Ayad@lmpa.univ-littoral.fr}\\
\ \\
Vincenzo Coia\\
Department of Statistics\\
University of British Columbia\\
Vancouver, BC  V6T 1Z4\\
Canada\\
\href{mailto:vincen.coia@stat.ubc.ca}{\tt vincen.coia@stat.ubc.ca} \\
\ \\
Omar Kihel \\
Department of Mathematics \\
Brock University\\
St.\ Catharines, ON  L2S 3A1\\
Canada\\
\href{mailto:okihel@brocku.ca}{\tt okihel@brocku.ca} \\
\end{center}

\vskip .2 in


\begin{abstract}
Let $A$ be a finite union of disjoint sets of consecutive integers
and let $n$ be a positive integer. We give a formula for the number of relatively
prime subsets (resp., relatively prime subsets of cardinality $k$)
of $A$, which generalizes results of Nathanson, El Bachraoui
and others. We give as well similar formulas for the number of subsets with gcd coprime to $n$. 
\end{abstract}

\section{Introduction}

A nonempty set $S$ of integers is said to be \emph{relatively prime}
if $\gcd\left(S\right)=1$, where $\gcd\left(S\right)$ denotes the
greatest common divisor of the elements of $S$. Nathanson \cite{Nat1}
defines $f\left(n\right)$ to be the number of relatively prime subsets
of $\left\{ 1,2,\ldots,n\right\} $ and, for $k\geq1$, $f_{k}\left(n\right)$
to be the number of relatively prime subsets of $\left\{ 1,2,\ldots,n\right\} $
of cardinality $k$. By analogy with Euler's phi function $\phi\left(n\right)$
that counts the number of positive integers $a$ in the set $\left\{ 1,2,\ldots,n\right\} $
such that $\gcd\left(a,n\right)=1$, Nathanson \cite{Nat1} defines
$\Phi\left(n\right)$ to be the number of nonempty subsets $S$ of
the set $\left\{ 1,2,\ldots,n\right\} $ such that $\gcd\left(S\right)$
is relatively prime to $n$ and, for integer $k\geq1$, $\Phi_{k}\left(n\right)$
to be the number of subsets $S$ of the set $\left\{ 1,2,\ldots,n\right\} $
such that $\gcd(S)$ is relatively prime to $n$ and $\left|S\right|=k$.
He obtains explicit formulas for these four functions and deduces
asymptotic estimates.

For simplicity, we use a more general notation than Nathanson \cite{Nat1}.
For a nonempty set of integers $S$, we define
\begin{itemize}
\item $f\left(S\right)=\left|\left\{ H\subseteq S:\gcd\left(H\right)=1,H\neq\emptyset\right\} \right|$
as the number of nonempty relatively prime subsets of $S$;
\item $f_{k}\left(S\right)=\left|\left\{ H\subseteq S:\gcd\left(H\right)=1,\left|H\right|=k\right\} \right|$
as the number of relatively prime subsets of $S$ of cardinality $k$;
\item $\Phi\left(S,n\right)=\left|\left\{ H\subseteq S:\gcd\left(H\cup\left\{ n\right\} \right)=1,H\neq\emptyset\right\} \right|$
as the number of nonempty subsets of $S$ with $\gcd$ relatively
prime to integer $n$; 
\item $\Phi_{k}\left(S,n\right)=\left|\left\{ H\subseteq S:\gcd\left(H\cup\left\{ n\right\} \right)=1,\left|H\right|=k\right\} \right|$
as the number of subsets of $S$ of cardinality $k$ and with $\gcd$
relatively prime to integer $n$.
\end{itemize}
Further, we define $\left[a,b\right]_{\mathbb{Z}}=\left[a,b\right]\cap\mathbb{Z}=\left\{ a,a+1,\ldots,b\right\} $
for integers $a&lt;b$ as the set of consecutive integers from $a$ to
$b$, inclusive.

El Bachraoui \cite{Bach1} and Nathanson and Orosz \cite{Nat} generalize
the results of Nathanson \cite{Nat1} to subsets of $\left[\ell,m\right]_{\mathbb{Z}}$
for integers $0\leq\ell&lt;m$, and prove Theorem \ref{Thm:_f,_fk_for_consecutive}.

\begin{theorem} \label{Thm:_f,_fk_for_consecutive}For non-negative integers
$\ell&lt;m$ and $k\geq1$, using the notation $f\left(\ell,m\right)=f\left(\left[\ell,m\right]_{\mathbb{Z}}\right)$
and $f_{k}\left(\ell,m\right)=f_{k}\left(\left[\ell,m\right]_{\mathbb{Z}}\right)$
of El Bachraoui \cite{Bach1} and Nathanson and Orosz \cite{Nat}
we have 
\begin{equation}
f\left(\ell,m\right)=\sum_{d=1}^{m}\mu\left(d\right)\left(2^{\left\lfloor \frac{m}{d}\right\rfloor -\left\lfloor \frac{\ell}{d}\right\rfloor }-1\right)
\end{equation}
and 
\begin{equation}
f_{k}\left(\ell,m\right)=\sum_{d=1}^{m}\mu\left(d\right){
\left\lfloor m/d\right\rfloor -\left\lfloor \ell/d\right\rfloor \choose k},
\end{equation}
where $\mu$ is the M\"{o}bius function.\end{theorem} 

For brevity, define the arithmetic sequence $\mathcal{A}_{n}^{\left(a,b\right)}=\left\{ a,a+b,\ldots,a+(n-1)b\right\} $
for positive integers $n$, $a$, and $b$. Ayad and Kihel \cite{Aya1}
generalize Theorem \ref{Thm:_f,_fk_for_consecutive} to obtain Theorem
\ref{Thm:_f,_fk_for_arithmetic}. 

\begin{theorem} \label{Thm:_f,_fk_for_arithmetic}For all positive integers
$n$, $a$, and $b$, with $\gcd\left(a,b\right)=1$, using the notation
$f^{\left(a,b\right)}\left(n\right)=f\left(\mathcal{A}_{n}^{\left(a,b\right)}\right)$
and $f_{k}^{\left(a,b\right)}\left(n\right)=f_{k}\left(\mathcal{A}_{n}^{\left(a,b\right)}\right)$
of Ayad and Kihel \cite{Aya1}, we have 
\begin{equation}
f^{\left(a,b\right)}\left(n\right)=\sum_{\substack{d=1\\
\gcd\left(b,d\right)=1
}
}^{a+\left(n-1\right)b}\mu\left(d\right)\left(2^{\left\lfloor \frac{n}{d}\right\rfloor +\varepsilon_{d}}-1\right)
\end{equation}
and 
\begin{equation}
f_{k}^{\left(a,b\right)}\left(n\right)=\sum_{\substack{d=1\\
\gcd\left(b,d\right)=1
}
}^{a+\left(n-1\right)b}\mu\left(d\right){
\left\lfloor \frac{n}{d}\right\rfloor +\varepsilon_{d} \choose k},
\end{equation}
where 
\begin{equation}
\varepsilon_{d}=\begin{cases}
0, &amp; \text{if }d\mid n;\\
1, &amp; \text{if }d\nmid n\text{ and }\left(-ab^{-1}\right)\bmod d\in\left\{ \left\lfloor \frac{n}{d}\right\rfloor d,\ldots,n-1\right\} ;\\
0, &amp; \text{otherwise}.
\end{cases}
\end{equation}
\end{theorem} 

El Bachraoui \cite{Bach3} extends Theorem \ref{Thm:_f,_fk_for_consecutive}
to the union of two sets of consecutive integers, to obtain Theorem
\ref{Thm:_f,_fk_for_union_of_two_consecutive_sets}. 

\begin{theorem} \label{Thm:_f,_fk_for_union_of_two_consecutive_sets}For
nonnegative integers $\ell_{1}&lt;m_{1}&lt;\ell_{2}&lt;m_{2}$ and for $k\geq1$,
\begin{equation}
f\left(\left[\ell_{1},m_{1}\right]_{\mathbb{Z}}\cup\left[\ell_{2},m_{2}\right]_{\mathbb{Z}}\right)=\sum_{d=1}^{m_{2}}\mu\left(d\right)\left(2^{\left\lfloor \frac{m_{1}}{d}\right\rfloor +\left\lfloor \frac{m_{2}}{d}\right\rfloor -\left\lfloor \frac{\ell_{1}-1}{d}\right\rfloor -\left\lfloor \frac{\ell_{2}-1}{d}\right\rfloor }-1\right)
\end{equation}
and 
\begin{equation}
f_{k}\left(\left[\ell_{1},m_{1}\right]_{\mathbb{Z}}\cup\left[\ell_{2},m_{2}\right]_{\mathbb{Z}}\right)=\sum_{d=1}^{m_{2}}\mu\left(d\right){
\left\lfloor \frac{m_{1}}{d}\right\rfloor +\left\lfloor \frac{m_{2}}{d}\right\rfloor -\left\lfloor \frac{\ell_{1}-1}{d}\right\rfloor -\left\lfloor \frac{\ell_{2}-1}{d}\right\rfloor \choose k}.
\end{equation}
\end{theorem} 

We now switch our attention to analogous results for functions $\Phi$
and $\Phi_{k}$. For the consecutive integers case, El Bachraoui \cite{Bach1}
and Nathanson and Orosz \cite{Nat} prove Theorem \ref{Thm:_Phi,_Phi_k_for_consecutive}.

\begin{theorem} \label{Thm:_Phi,_Phi_k_for_consecutive}For non-negative
integers $\ell&lt;m$ and $k\geq1$, using the notation $\Phi\left(\ell,m\right)=\Phi\left(\left[\ell,m\right]_{\mathbb{Z}},m\right)$
and $\Phi_{k}\left(\ell,m\right)=\Phi_{k}\left(\left[\ell,m\right]_{\mathbb{Z}},m\right)$
of El Bachraoui \cite{Bach1} and Nathanson and Orosz \cite{Nat}
we have 
\begin{equation}
\Phi\left(\ell,m\right)=\sum_{d\mid m}\mu\left(d\right)2^{\left(\frac{m}{d}-\left\lfloor \frac{\ell}{d}\right\rfloor \right)}
\end{equation}
and 
\begin{equation}
\Phi_{k}\left(\ell,m\right)=\sum_{d\mid m}\mu\left(d\right){\frac{m}{d}-\left\lfloor \frac{\ell}{d}\right\rfloor  \choose k}.
\end{equation}
\end{theorem}

Ayad and Kihel \cite{Aya1} generalize Theorem \ref{Thm:_Phi,_Phi_k_for_consecutive}
to obtain Theorem \ref{Thm:_Phi,_Phi_k_for_arithmetic}. 

\begin{theorem}\label{Thm:_Phi,_Phi_k_for_arithmetic} For nonnegative
integers $a$, $b$, and $n$, with $\gcd\left(a,b\right)=1$, using
the notation $\Phi^{\left(a,b\right)}\left(n\right)=\Phi\left(\mathcal{A}_{n}^{\left(a,b\right)},n\right)$
and $\Phi_{k}^{\left(a,b\right)}\left(n\right)=\Phi_{k}\left(\mathcal{A}_{n}^{\left(a,b\right)},n\right)$
of Ayad and Kihel \cite{Aya1} we have 
\begin{equation}
\Phi^{\left(a,b\right)}\left(n\right)=\sum_{\substack{d\mid n\\
\gcd\left(b,d\right)=1
}
}\mu\left(d\right)\left(2^{\frac{n}{d}}-1\right)
\end{equation}
and 
\begin{equation}
\Phi_{k}^{\left(a,b\right)}\left(n\right)=\sum_{\substack{d\mid n\\
\gcd\left(b,d\right)=1
}
}\mu\left(d\right){\frac{n}{d} \choose k}.
\end{equation}
\end{theorem}

El Bachraoui and Salim \cite{Bach5} extend Theorem \ref{Thm:_Phi,_Phi_k_for_consecutive}
to the union of two sets of consecutive integers, to obtain Theorem
\ref{Thm:_Phi,_Phi_k_for_union_of_two_consecutive_sets}. 

\begin{theorem} \label{Thm:_Phi,_Phi_k_for_union_of_two_consecutive_sets}For
nonnegative integers $\ell_{1}&lt;m_{1}&lt;\ell_{2}&lt;m_{2}$ and for $k\geq1$,
\begin{equation}
\Phi\left(\left[\ell_{1},m_{1}\right]_{\mathbb{Z}}\cup\left[\ell_{2},m_{2}\right]_{\mathbb{Z}},n\right)=\sum_{d\mid n}\mu\left(d\right)2^{\left\lfloor \frac{m_{1}}{d}\right\rfloor +\left\lfloor \frac{m_{2}}{d}\right\rfloor -\left\lfloor \frac{\ell_{1}-1}{d}\right\rfloor -\left\lfloor \frac{\ell_{2}-1}{d}\right\rfloor }
\end{equation}
and 
\begin{equation}
\Phi_{k}\left(\left[\ell_{1},m_{1}\right]_{\mathbb{Z}}\cup\left[\ell_{2},m_{2}\right]_{\mathbb{Z}},n\right)=\sum_{d\mid n}\mu\left(d\right){\left\lfloor \frac{m_{1}}{d}\right\rfloor +\left\lfloor \frac{m_{2}}{d}\right\rfloor -\left\lfloor \frac{\ell_{1}-1}{d}\right\rfloor -\left\lfloor \frac{\ell_{2}-1}{d}\right\rfloor  \choose k}.
\end{equation}
\end{theorem} 

In Section \ref{Sec:_our_contribution}, we extend Theorems \ref{Thm:_f,_fk_for_union_of_two_consecutive_sets}
and \ref{Thm:_Phi,_Phi_k_for_union_of_two_consecutive_sets} to the
union of any finite number of disjoint sets of consecutive integers. The approach we take is simple and much different from the approach of El Bachraoui \cite{Bach3}
and El Bachraoui and Salim \cite{Bach5} for the union of two sets.
Several authors \cite{Aya2,Aya3,Shonh,Bach45,Tan,La,Bach2} discuss other properties and generalizations. 



\section{Finite union of disjoint sets of consecutive integers\label{Sec:_our_contribution}}

For positive integers $\ell_{i}\leq m_{i}$ for $i=1,\ldots,r$,
denote $A^{\left(i\right)}=\left[\ell_{i},m_{i}\right]_{\mathbb{Z}}$
for brevity and assume $A^{(i)}\cap A^{(j)}=\emptyset$ for $i\neq j$.
Consider the union 
\begin{equation}
A=\overset{r}{\underset{i=1}{\bigcup}}A^{(i)}.\label{eq:Defn_of_A}
\end{equation}
El Bachraoui \cite{Bach3} derives equations for $f\left(A\right)$
and $f_{k}\left(A\right)$ for $r=2$, as in Theorem \ref{Thm:_f,_fk_for_union_of_two_consecutive_sets}.
We extend this to any $r\in\mathbb{N}$ in Theorem \ref{Thm-New_f},
but first we need Lemma \ref{LEM:_|Ad|_in_union}. Also, throughout
this section, for a set of integers $S$ we denote $\mathcal{P}\left(S\right)=\left\{ H\subseteq S:H\neq\emptyset\right\} $
and $\mathcal{P}_{k}\left(S\right)=\left\{ H\subseteq S:\left|H\right|=k\right\} $.

 \begin{lemma}\label{LEM:_|Ad|_in_union}Let $A_{d}=\left\{ x\in A:d\mid x\right\} $
be all the multiples of $d$ found in $A$, where $A$ is defined
in equation \eqref{eq:Defn_of_A}. Then, 
\[
\left|A_{d}\right|=\overset{r}{\underset{i=1}{\sum}}\left(\left\lfloor \frac{m_{i}}{d}\right\rfloor -\left\lfloor \frac{\ell_{i}-1}{d}\right\rfloor \right).
\]


\end{lemma}

\begin{proof} For $i=1,\ldots,r$, let $A_{d}^{(i)}=\left\{ x\in A^{\left(i\right)}:d\mid x\right\} $,
$M_{d}^{(i)}=\left\{ x\in\left[0,m_{i}\right]_{\mathbb{Z}}:d\mid x\right\} $,
and $L_{d}^{(i)}=\{x\in\left[0,\ell_{i}-1\right]_{\mathbb{Z}}:d\mid x\}$.
Clearly, we have $\left|A_{d}^{(i)}\right|=\left|M_{d}^{(i)}\right|-\left|L_{d}^{(i)}\right|$.
But, we simply have $\left|M_{d}^{(i)}\right|=\left\lfloor \frac{m_{i}}{d}\right\rfloor $
and $\left|L_{d}^{(i)}\right|=\left\lfloor \frac{\ell_{i}-1}{d}\right\rfloor $.
So, 
\[
\left|A_{d}^{(i)}\right|=\left\lfloor \frac{m_{i}}{d}\right\rfloor -\left\lfloor \frac{\ell_{i}-1}{d}\right\rfloor .
\]
Now, since $A_{d}=\overset{r}{\underset{i=1}{\bigcup}}A_{d}^{(i)}$
and since $A_{d}^{(i)}\cap A_{d}^{(j)}=\emptyset$ for $i\neq j$,
we have $\left|A_{d}\right|=\overset{r}{\underset{i=1}{\sum}}\left|A_{d}^{(i)}\right|$
which completes the proof. \end{proof}

 \begin{theorem}\label{Thm-New_f}For $A$ defined in equation
\eqref{eq:Defn_of_A}, we have 
\begin{equation}
f(A)=\overset{\max\left\{ m_{1},\ldots,m_{r}\right\} }{\underset{d=1}{\sum}}\mu\left(d\right)\left(2^{\overset{r}{\underset{i=1}{\sum}}\left(\left\lfloor \frac{m_{i}}{d}\right\rfloor -\left\lfloor \frac{\ell_{i}-1}{d}\right\rfloor \right)}-1\right);\label{eq:f_for_union}
\end{equation}
\begin{equation}
f_{k}(A)=\overset{\max\left\{ m_{1},\ldots,m_{r}\right\} }{\underset{d=1}{\sum}}\mu\left(d\right){\overset{r}{\underset{i=1}{\sum}}\left(\left\lfloor \frac{m_{i}}{d}\right\rfloor -\left\lfloor \frac{\ell_{i}-1}{d}\right\rfloor \right) \choose k}.\label{eq:fk_for_union}
\end{equation}


\end{theorem} 

\begin{proof}\textbf{ }We begin by proving equation \eqref{eq:f_for_union}.
From the total amount of nonempty subsets of $A$, remove those subsets
that are \emph{not} relatively prime: 
\[
f\left(A\right)=\left|\mathcal{P}\left(A\right)\right|-\left|\underset{p\text{ prime}}{\bigcup}\mathcal{P}\left(A_{p}\right)\right|.
\]
Using inclusion-exclusion and the same principle as in the proof of
Ayad and Kihel \cite[Theorem 5]{Aya1}, we obtain 
\[
f\left(A\right)=\overset{\max\left\{ m_{1},\ldots,m_{r}\right\} }{\underset{d=1}{\sum}}\mu\left(d\right)\left(2^{\left|A_{d}\right|}-1\right).
\]
Applying Lemma \eqref{LEM:_|Ad|_in_union}, we obtain equation \eqref{eq:f_for_union}.

To prove equation \eqref{eq:fk_for_union}, from the total amount
of subsets of $A$ with cardinality $k$, remove those subsets that
are \emph{not} relatively prime: 
\[
f_{k}\left(A\right)=\left|\mathcal{P}_{k}\left(A\right)\right|-\left|\underset{p\text{ prime}}{\bigcup}\mathcal{P}_{k}\left(A_{p}\right)\right|.
\]
Using inclusion-exclusion and the same principle as in the proof of
Ayad and Kihel \cite[Theorem 5]{Aya1}, we obtain 
\[
f_{k}(A)=\overset{\max\left\{ m_{1},\ldots,m_{r}\right\} }{\underset{d=1}{\sum}}\mu\left(d\right)\binom{\left|A_{d}\right|}{k}.
\]
Applying Lemma \eqref{LEM:_|Ad|_in_union}, we obtain equation \eqref{eq:fk_for_union}.
\end{proof}

Similarly, we now extend Theorem \ref{Thm:_Phi,_Phi_k_for_union_of_two_consecutive_sets}.

\begin{theorem}\label{Thm-New_Phi}Define $A$ as in equation \eqref{eq:Defn_of_A}.
Then for any integer $k\geq1$, 
\begin{equation}
\Phi\left(A,n\right)=\underset{d\mid n}{\sum}\mu\left(d\right)\left(2^{\overset{r}{\underset{i=1}{\sum}}\left(\left\lfloor \frac{m_{i}}{d}\right\rfloor -\left\lfloor \frac{\ell_{i}-1}{d}\right\rfloor \right)}-1\right);\label{eq:Phi_for_union}
\end{equation}


\begin{equation}
\Phi_{k}\left(A,n\right)=\underset{d\mid n}{\sum}\mu\left(d\right){\overset{r}{\underset{i=1}{\sum}}\left(\left\lfloor \frac{m_{i}}{d}\right\rfloor -\left\lfloor \frac{\ell_{i}-1}{d}\right\rfloor \right) \choose k}.\label{eq:Phi_k_for_Union}
\end{equation}


\end{theorem}

\begin{proof} We begin by proving equation \eqref{eq:Phi_for_union}.
Notice that 
\[
\Phi\left(A,n\right)=\left|\mathcal{P}\left(A\right)\right|-\left|\bigcup_{\substack{p\text{ prime}\\
p\mid n
}
}\mathcal{P}\left(A_{p}\right)\right|.
\]
As in the proof of Theorem \ref{Thm-New_f}, we have 
\[
\Phi\left(A,n\right)=\underset{d\mid n}{\sum}\mu\left(d\right)\left(2^{\left|A_{d}\right|}-1\right).
\]
Applying Lemma \ref{LEM:_|Ad|_in_union} proves equation \eqref{eq:Phi_for_union}. 

To prove equation \eqref{eq:Phi_k_for_Union}, notice that
\[
\Phi_{k}\left(A,n\right)=\left|\mathcal{P}_{k}\left(A\right)\right|-\left|\bigcup_{\substack{p\text{ prime}\\
p\mid n
}
}\mathcal{P}_{k}\left(A_{p}\right)\right|.
\]
 As in the proof of Theorem \ref{Thm-New_f}, we have 
\[
\Phi_{k}\left(A,n\right)=\underset{d\mid n}{\sum}\mu\left(d\right){\left|A_{d}\right| \choose k}.
\]
Applying Lemma \ref{LEM:_|Ad|_in_union} proves equation \eqref{eq:Phi_k_for_Union}.\end{proof} 

\section{Aknowledgements} The authors would like to thank the anonymous referees for several helpful comments. The research of the third author is patially supported by NSERC.

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\bibitem{La}L.\ T\'{o}th, \textit{\emph{On the number of
certain relatively prime subsets of $\{1,2,\ldots,n\}$}}, \emph{Integers}
\textbf{10} (2010), 407--421.
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\noindent 2010 {\it Mathematics Subject Classification}:
Primary 11A25; Secondary 11B05, 11B75, 11D41.

\noindent \emph{Keywords: } 
phi function, relatively prime set,  
combinatorial identity.

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\noindent
Received November 10 2011;
revised versions received  November 11 2011; June 10 2013; September 17 2013; 
January 27 2014.
Published in {\it Journal of Integer Sequences}, February 16 2014.

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