The Number of r-Relatively Prime Subsets and Phi Functions for {m,m+1, …, n}- its r-Generalization

Vol. 14, No. 3, 2017, 581-592

ISSN: 2279-087X (P), 2279-0888(online) Published on 5 December 2017

www.researchmathsci.org

DOI: http://dx.doi.org/10.22457/apam.v14n3a25

581

Annals of

The Number of r-Relatively Prime Subsets and Phi

Functions for {m,m+1, …, n}- its r-Generalization

G. Kamala1 and G. Lalitha2

1

Department of Mathematics, Osmania University, Hyderabad (TS), India

2

Department of Mathematics, SRR & CVR Govt. Degree College Vijayawada (AP), India.

Email: 1gkamala369@gmail.com;2lalitha.secpg@gmail.com

2

Corresponding author.

Received 15 November 2017; accepted 30 November 2017

Abstract. A nonempty finite set A of positive integers is relatively prime if gcd

( )

A =1 and it is relatively prime to n ifgcd

(

A∪

{ }

n

)

=1. For k≥1, the number of nonempty subsets A _{of {1,2,…,n} which are relatively prime is} f n

( )

and the number of such subsets of cardinality k, is f_k

( )

n . The number of nonempty subsets A of {1,2,…,n} which are relatively prime to n is Φ

( )

n and the number of such subsets of cardinality k isΦ_k

( )

n . A nonempty finite set A of positive integers is r-relatively prime if greatest r th power common divisor of elements of A is 1. In this case we writegcd_r

( )

A =1. The number of r-relatively prime subsets of {1,2,…,n} is f

( )

r

( )

n and the number of such subsets of cardinality k is f_k

( )

r

( )

n . The number of nonempty subsets A _{of {1,2,…,n}} which are r-relatively prime to n is Φ

( )

r

( )

n and the number of such subsets of cardinality k is Φ

( )

_kr

( )

n . In this paper we generalize the four functions f

( )

r

( )

n ,

( )

r

_{( )}

k

f n ,Φ

( )

r

( )

n and Φ

( )

_kr

( )

n for the set

{

m m, +1, ..., n

}

where m≤n, as follows :

( )

r

₍

_,

₎

_#

_{

_{

_{, 1, ...,}

_}

_{: , gcd}

_{( )}

₁

_}

r

f m n = A⊆ m m+ n A≠

φ

A =

( )

r

₍

_,

₎

_#

_{

_{

_{, 1, ...,}

_}

_{: # , gcd}

_{( )}

₁

_}

r k

f m n = A⊆ m m+ n A=k A =

582

Φ

( )

_kr

(

m n,

)

=#

{

A⊆

{

m m, +1, ..., n

}

: #A=k , gcd_r

(

A∪

{ }

n

)

=1 .

}

We obtain the exact formulae for the above four functions. We use r-Generalized Mobius inversion formula for two variables.

Keywords: r-relatively prime sets, r-Generalization of Phi functions, r-Generalized Mobius inversion for two variables.

AMS Mathematics Subject Classification (2010): 11BXX, 11B75 1. Introduction

A nonempty finite set A of positive integers is relatively prime if gcd

( )

A =1 and it is relatively prime to n ifgcd

(

A∪

{ }

n

)

=1. The number of nonempty subsets of {1,2,…,n} which are relatively prime is f n

( )

and the number of such subsets of cardinality k is f_k

( )

n . The number of nonempty subsets A _{of {1, 2 ,…, n} which are} relatively prime to n is Φ

( )

n and the number of such subsets of cardinality k is

( )

.

k n

Φ A nonempty finite set A of positive integers {1, 2 ,…, n} is r-relatively prime if greatest rth power common divisor of elements of A is 1. In this case, we write

( )

gcd_r A =1. The number of r-relatively prime subsets of {1, 2,…, n} is f

( )

r

( )

n and the number of such subsets of cardinality k is f_k

( )

r

( )

n . The number of nonempty subsets A of {1, 2 ,…, n} which are r-relatively prime to n is Φ

( )

r

( )

n and the number of such subsets of cardinality k is Φ_k

( )

r

( )

n . Nathanson [2] obtained the exact formulae for the functions f n

( )

, f_k

( )

n ,Φ

( )

n and Φ_k

( )

n . Inspired by the work of Nathanson [2] , we obtained the exact formulae for f

( )

r

( )

n , f_k

( )

r

( )

n ,Φ( )r

( )

n and Φ( )_kr

( )

n in [1]. Bachraoui [3] generalized the results of Nathanson [2]. In this paper we generalize the results obtained in [1] and derive the explicit formulae for the functions f

( )

r

(

m n,

)

,

( )

r

₍

_,

₎

k

f m n ,Φ

( )

r

(

m n,

)

and Φ

( )

_kr

(

m n,

)

by using r-Generalized Mobius inversion formula for two variables.

Definition 1. Let r ≥ _{1 be a positive integer. F}

( )

r _{and G}

( )

r _{are generalized arithmetical} functions of two variables with domain

(

0, ∞

)

2 such that

( )

₍

₎

( )

₍

₎

1, 2 0 1, 2

r r

{m,m+1, …, n}- its r -Generalization

583

Definition 2. r - Relatively prime subsets of {m, m+1, …, n}. Let r≥1, k≥1 be fixed integers .Then we define,

g

( )

r

(

m n,

)

=#

{

A⊆

{

m m, +1, ..., n

}

: m∈A, gcd_r

( )

A =1

}

g_k

( )

r

(

m n,

)

=#

{

A⊆

{

m m, +1, ..., n

}

: m∈A, #A=k, gcd_r

( )

A =1

}

f

( )

r

(

m n,

)

=#

{

A⊆

{

m m, +1, ..., n

}

: A≠

φ

, gcd_r

( )

A =1

}

f_k

( )

r

(

m n,

)

=#

{

A⊆

{

m m, +1, ..., n

}

: A≠

φ

, #A=k, gcd_r

( )

A =1 .

}

Note that f

( )

r , f_k

( )

r , g

( ) ( )

r , g_kr are counting sets on nonempty subsets.

( )

r

_{( )}

f n = Number of r - relatively prime subsets of

{

1, 2, ..., n

}

= f

( )

r

( )

1, n . Definition 3. Let

n

≥

m

be positive integers. We define

( )

r

₍

_,

₎

_#

_{

_{

_{, 1, ...,}

_}

_{: , gcd}

₍

_{{ }}

₎

₁

_}

r

m n A m m n m A A n

ψ

= ⊆ + ∈ ∪ =

( )

r

₍

_,

₎

_#

_{

_{

_{, 1, ...,}

_}

_{: , # , gcd}

₍

_{{ }}

₎

₁

_}

r

k m n A m m n m A A k A n

ψ

= ⊆ + ∈ = ∪ =

( )

r

₍

_,

₎

_#

_{

_{

_{, 1, ...,}

_}

_{: , gcd}

₍

_{{ }}

₎

₁

_}

r

m n A m m n A

φ

A n

Φ = ⊆ + ≠ ∪ =

( )

r

₍

_,

₎

_#

_{

_{

_{, 1, ...,}

_}

_{: # , gcd}

₍

_{{ }}

₎

_{1 .}

_}

r

k m n A m m n A k A n

Φ = ⊆ + = ∪ =

Note that the four functions defined above count the sets that are nonempty.

( )

r

_{( )}

n

Φ =Number of non empty subsets of

{

1, 2, ..., n

}

that are r-relatively prime to n = Φ

( )

r

( )

1,n

2. r-Generalized Mobius inversion formula for two variables

Theorem 1. Let F

( )

r and G

( )

r be generalized arithmetical functions with domain

(

)

2

0, ∞ then

( )

₍

₎

( ) ( )

( )

( )

1 1

, , , ,

r r r r r r

r r

r r m n r m n r r m n

r

d d d d d d

d m d m

G m n F F µ d G

≤ ≤ ≤ ≤

     

= ∑ _{ } ⇔ _{ }= ∑ _{ }

      (1)

where the summation is over all integers d such that dr ≤m. Proof: Assume

( )

(

)

( )

1

, _r, _r

r

r r _{m n}

d d d m

G m n F

≤ ≤

 

= ∑ _{ }

584 Consider

( )

( )

( )

( )

1 1 1

, ,

r

m

d

r r r r r r

r r r

r r

r m n r m n

r r

d d t d t d

d m d m t

d G d F

µ

µ

≤ ≤ ≤ ≤ ≤ ≤

 

 

  ₌  

∑ _{ } ∑ _ ∑ _{ }

  _  _

 

( )

( )

1

,

r r

r r r

r m n r

r

u u

u m d u

F

µ

d

≤ ≤

 

 _{  }

= ∑ _{ } ∑

   

 

=F

( )

r

(

m n,

)

.

Conversely, assume

( )

(

)

( )

( )

1

, , .

r r

r

r r r m n

r

d d d m

F m n

µ

d G

≤ ≤

 

= ∑ _{ }

 

Consider

( )

( )

( )

1 1 1

, ,

r

m d

r r r r r r

r r r

r m n r r m n

r

d d t d t d

d m d m t

F

µ

d G

≤ ≤ ≤ ≤ ≤ ≤

 

 

  ₌  

∑ _{ } ∑ _ ∑ _{ }

  _  _

 

( )

( )

1

,

r r

r r r

r m n r

r

u u

u m t u

G

µ

t

≤ ≤

 

 _{  }

= ∑ _{ } ∑

   

 

=G

( )

r

(

m n,

)

.

Corollary 2. Suppose F

( )

r

(

x y,

)

and G

( )

r

(

x y,

)

are generalized arithmetical functions such that F

( )

r

(

x y,

)

= =0 G

( )

r

(

x y,

)

if x is not an integer and

( )

r

₍

_,

₎

₀

( )

r

₍

_,

₎

F x y = =G x y if 0< <y 1, then Mobius inversion formula for such functions reduces to the following form

( )

₍

_,

₎

( )

_,

( )

₍

_,

₎

( )

_,

r r r r

r r

r r m n r r m n

r

d d d d

d m d m

G m n = ∑ F _ _ __⇔F m n = ∑

µ

d _{ } _

   

    (2)

Proof: Suppose

( )

(

,

)

( )

,

r r

r

r r _{m n}

d d

d m

G m n = ∑ F _ _ __

 

{m,m+1, …, n}- its r -Generalization

585 Consider

( )

( )

_,

( )

( )

_,

r r r r r r

r r r m

r d

r r

r m n r m n

r r

d d t d t d

d m d m t

d G d F

µ

µ

 

 

  ₌   

∑ _{  } ∑  ∑ _{  }

   

  _  _

 

 

( )

,

( )

r r

r r r

r m n r

r

u u

u m d u

F

µ

d

 

 

  

= ∑ _{  } ∑ _

 

 _{ }

 

=F

( )

r

(

m n,

)

.

Conversely, suppose

( )

(

,

)

( )

( )

_r , _r

r

r r r m n

r

d d

d m

F m n = ∑

µ

d G _ _ __

 

 

Consider

( )

_r, _r

r

r _{m n}

d d

d m

F   

∑ _{  }

 

 

( )

( )

_,

r r r r

r r m

r d

r

r m n

r

t d t d

d m t

t G

µ

 

 _{  }

= ∑  ∑ _{  }

 

 

 

 

 

( )

_r, _r

( )

r r r

r m n r

r

u u

u m t u

G

µ

t

 

 

  

= ∑ _{   } ∑ _

 

 _{  }

 

=G

( )

r

(

m n,

)

. 3. Exact formulae

To find the exact formulas we use the following theorems proved in [1]. Theorem 3. For all positive integers n≥2r r≥1, k,

(i)

( )

( )

( )

( )

( )

1

2 1 1,

n r d r

r r r

r

d n

f n

µ

d f n

     

≤ ≤

 

 

= ∑  − =

 

 

(3)

(ii)

( )

( )

( )

( )

( )

1

1, .

n r d r

r _r r

r

k k

d n

f n d f n

k

µ

 

 

 

≤ ≤

 

 

= ∑ _{ }=

 

 

(4)

586

(i)

( )

( )

( )

2 1

( )

( )

1, .

n r d r

r r r r

d n

n

µ

d n

 

 

Φ = ∑ − = Φ

 

 

(5)

(ii)

( )

_k

( )

( )

r

( )

_k

( )

1, .

r

n

r r r _d r

d n

n d n

k

µ

 

Φ = ∑ = Φ

 

 

(6)

Lemma 5. If n≥m, we have

(i)

( )

(

,

)

( )

2

n m

r r

d d

r

r r

r d m

g m n

µ

d

  −    

= ∑ (7)

(ii)

( )

(

,

)

( )

.

1

r r

r

n m

r r

d d

r k

d m

g m n d

k

µ

 ₋ 

  

= ∑ _  _

 ₋ 

 

(8)

Proof: (i) Let P m n

(

,

)

denotes the set of subsets of

{

m m, +1, ..., n

}

containing m. Then# P m n

(

,

)

=2n m− . We define a relation on P m n

(

,

)

as follows:

For A B, ∈P m n

(

,

)

we say that A≅ B if and only if gcd_r

( )

A =dr =gcd_r

( )

B . Note that d mr and 1≤dr ≤m. Then

≅

is an equivalence relation on P m n

(

,

)

. If m is r-free then gcd_r

( )

A = =1 gcd_r

( )

B . Hence the elements of P m n

(

,

)

can be partitioned using the relation

" "

≅

of having the same rth power gcd dr. Also the mapping A 1_r A

d

→ is a 1-1 correspondence between the subsets of P m n

(

,

)

having

th

r power gcd dr and r-relatively prime subsets of , ...,

r r

m n

d d

 

 

  which contain

.

r

m

d Then we have the following identity

( )

2 , .

r r

r

r

n m m n

d d

d m

g

− _{= ∑}   

 _ _

 

By Corollary (2)

( )

(

,

)

( )

2

n m

r r

d d

r

r r

r d m

g m n

µ

d

 

−

 

 

= ∑ which proves (7).

(ii) Note that the correspondence 1

r

A A

d

{m,m+1, …, n}- its r -Generalization

587

( )

, .

1 r r r

r _{m n}

k d d d m n m g k −  ₌    ∑  ₋   _{ }    

By Corollary (2), we have

( )

(

,

)

( )

. 1 r r r n m r r d d r k d m

g m n d

k

µ

 ₋     = ∑ _  _  ₋   

which proves (8).

Theorem 6. If n≥m, then the following two identities are true.

(i)

( )

(

)

( )

( )

1

1 1

, 2 1 2

n n i

r r r

d d d

r r

m

r r r

r r

i

d n d i

f m n

µ

d

µ

d

    −   ₋       = ≤ ≤     = ∑ _ − −_ ∑ ∑     (9)

(ii)

( )

(

)

( )

( )

1 1 1

, .

1

n n i

r r r

d d d

r r

m

r r r

r r

k

i

d n d i

f m n d d

k k

µ

µ

    − −         = ≤ ≤         = ∑ _{ }− ∑ ∑ _{ }    ₋      (10)

Proof: (i) Repeatedly applying Lemma (5) and equations (7) and (3)

f

( )

r

(

m n,

)

= f

( )

r

(

m−1, n

)

−g

( )

r

(

m−1, n

)

= f

( )

r

(

m−2, n

)

−_g

( )

r

(

m−2, n

)

+g

( )

r

(

m−1, n

)

_

  ⋮ ⋮

( )

( )

( )

1 1

1, m ,

r r

i

f n − g i n

= = − ∑

( )

( )

1 1 1

2 1 2

n n i

r r r

d d d

r r

m

r r

r r

i

d n d i

d d

µ

µ

    −   ₋       = ≤ ≤         = ∑  − − ∑  ∑         

(ii) From equation (4), we have

( )

( )

( )

1

1, .

n r d r r _r r k d n

f n d

k

µ

      ≤ ≤     = ∑ _{ }    

588

( )

(

,

)

( )

1

n m

r r

d d

r

r _r

r k

d m

g m n d

k

µ

 

−

 

 

 

 

= ∑ _{ }

 ₋ 

 

Therefore f_k

( )

r

(

m n,

)

= f_k

( )

r

(

m−1, n

)

−g_k

( )

r

(

m−1, n

)

= f_k

( )

r

(

m−2, n

)

−_g_k

( )

r

(

m−2, n

)

+g_k

( )

r

(

m−1, n

)

_

 

⋮ ⋮

( )

( )

( )

( )

1

1

1, m ,

r r

k k

i

f n − g i n

=

= − ∑

( )

( )

1 1 1

.

1

n n i

r r r

d d d

r r

m

r r

r r

i

d n d i

d d

k k

µ

µ

   

− −

   

   

= ≤ ≤

   

   

= ∑ _{ }− ∑ ∑ _{ }

   ₋ 

   

Note : gcd_r

(

{ }

)

1 gcd

(

_r

( )

,

)

1.

r

A∪ n = ⇔ A n =

Proof: Assume gcd_r

(

A∪

{ }

n

)

=1. Let

(

gcd_r

( )

,

)

r

r A n =d

⇒ dr n and dr gcd_r( )A ( since gcd_r

( )

A a for all a∈A) ⇒ dr n and dr a for all a∈A

⇒ dr gcd_r

(

A∪

{ }

n

)

⇒ dr1 ⇒ d =1. Conversely, let

(

gcd_r

( )

,

)

1

r

A n = and gcd_r

(

A∪

{ }

n

)

=dr ⇒ dr a for all a∈A and dr n

⇒ dr a and dr n for all a∈A ⇒ dr gcd_r( )A and dr n

r

(

gcd_r( ),

)

r

d A n

⇒

⇒ dr1 ⇒ d =1.

{m,m+1, …, n}- its r -Generalization 589

( )

₍

₎

( )

(

)

1 2 1 2 1 2 gcd ,

, _r , _r

r r

n n

r r

d d

d n n

G n n = ∑ F  

 

( )

₍

₎

( )

( )

(

)

1 2 1 2 1 2 gcd , , , r r r r n n

r r r

r

d d

d n n

F n n

µ

d G  

⇔ = ∑ _{ }

 

Proof: Assume

( )

(

)

( )

(

)

1 2 1 2 1 2 gcd , , , r r r r n n r r d d

d n n

G n n = ∑ F  

 

Consider

( )

( )

(

)

1 2 1 2 gcd , , r r r r n n r r r d d

d n n

d G

µ

  ∑ _{ }  

( )

( )

(

)

1 2 1 2 1 2

gcd , _{gcd ,}

,

r r r r

r _{r n n}

r _r r r d d n n r r r

t d t d

d n n _t

d F

µ

              = ∑ ∑ _{ }  _{ }    

( )

( )

(

)

1 2 1 2 gcd , , r r

r r r

r

n n

r r

r

u u

u n n d u

F

µ

d

      = ∑ _{ } ∑ _  _{ }  

=F

( )

r

(

n₁, n₂

)

.

Conversely, assume

( )

(

)

( )

( )

(

)

1 2 1 2 1 2 gcd , , , r r r r n n

r r r

r

d d

d n n

F n n = ∑

µ

d G  

  Consider

( )

(

)

( )

( )

(

)

1 2 1 2

1 2

1 2 1 2

gcd , gcd , _{gcd ,}

, ,

r r r r r r

r r _{r n n}

r r _r

r r d d

n n n n

r r r

r

d d t d t d

d n n d n n _t

F

µ

t G

            _{= }  _ ∑ _{ } ∑ ∑ _{ }          

( )

( )

(

)

1 2 1 2 gcd , , r r

r r r

r

n n

r r

r

u u

u n n t u

G

µ

t

      = ∑ _{ } ∑ _  _{ }  

( )

₍

₎

1 2

G r n , n .

=

590

(i)

( )

(

)

( )

(

)

gcd ,

, 2

n m r d r

r

r r

r

d m n

m n d

ψ

µ

−

= ∑ (11)

(ii)

( )

(

)

( )

(

)

gcd ,

, .

n m r d r

r

r r

r k

d m n

m n d

k

ψ

µ

−

 

 

= ∑ _{ }

 

 

(12)

Proof: (i) Let P m n

(

,

)

denotes the set of subsets of

{

m m, +1, ..., n

}

containing m. Then # P m n

(

,

)

=2n m− . We define the set

(

, , r

)

{

{

, 1, ...,

}

: and gcd_r

(

{ }

)

r

}

P m n d = A⊆ m m+ n m∈A A∪ n =d

The set P m n

(

,

)

can be partitioned using the equivalence relation

" "

≅

for having the same rthpower

gcd,

that is , A≅ ⇔B A B, ∈P m n d

(

, , r

)

for some

(

)

gcd , . r

r

d m n Further, the mapping 1

r

A A

d

→ is a

1 1

−

correspondence between

(

, , r

)

P m n d and the set of subsets B of , 1, 2, ...,

r r r r

m m m n

d d d d

 

+ +

 

  such that

r

m B d ∈

and gcd_r B n_r 1. d

  

∪  =

 

 

 

Then we have #

(

, , r

)

( )

r m_r, n_r

d d

P m n d =

ψ

 

 .

Thus we have

(

)

(

)

( )

(

)

gcd , gcd ,

2 # , , ,

r r

r r

r r

r

n m r m n

d d

d m n d m n

P m n d

ψ

− _{= =}  

∑ ∑ _{ }

 

( )

(

)

gcd ,

Therefore 2 , .

r r

r r

r

n m m n

d d

d m n

ψ

− ₌  

∑ _{ }

 

Applying Lemma (7) we have

( )

(

)

( )

(

)

gcd ,

, 2

n m r d r

r

r r

r

d m n

m n d

ψ

µ

−

= ∑ which proves (11).

(ii) Note that the correspondence 1

r

A A

d

→ defined above preserves the Cardinality and hence using an argument similar to the one in part (i) , we have

{m,m+1, …, n}- its r -Generalization

591

( )

(

)

gcd ,

,

1 r r r

r

r _{m n}

k

d d

d m n

n m

k

ψ

−

 ₌  

∑  

 ₋  _{ }

 

is equivalent to which proves (12). Theorem 9. If n≥m, then the following two identities are true.

(i)

( )

(

)

( )

( )

( )

1

1 _{gcd ,}

, 2 1 2

n n i

r r

d d

r r

r m

r r r

r r

i

d n d i n

m n

µ

d

µ

d

− −

=

 

 

Φ = ∑ − − ∑ ∑

 

 

(13)

(ii)

( )

(

)

( )

( )

( )

1

1 _{gcd ,}

, .

1

r r

r r

r

n n i

m

r r _d r _d

r r

k

i

d n d i n

m n d d

k k

µ

−

µ

−

=

   

   

Φ = ∑ − ∑ ∑

   ₋ 

   

(14)

Proof: (i) Applying theorem (8) repeatedly and using the equation (5) we have Φ

( )

r

(

m n,

)

= Φ

( )

r

(

m−1, n

)

−

ψ

( )

r

(

m−1, n

)

= Φ

( )

r

(

m−2, n

)

−_

ψ

( )

r

(

m−2, n

)

+

ψ

( )

r

(

m−1, n

)

_

 

⋮ ⋮

( )

( )

( )

( )

1

1

1, ,

m

r r

i

n −

ψ

i n

=

= Φ − ∑

( )

( )

( )

1

1 _{gcd ,}

2 1 2

n n i

r r

d d

r r

r

m

r r

r r

i

d n d i n

d d

µ

µ

− −

=

 

  _{ }

 

= ∑ − − ∑ _ ∑ _

  _{ }

  _{ }

which proves (13).

(ii) Applying theorem (8) repeatedly and using equation (6) Φ

( )

_kr

(

m n,

)

= Φ_k

( )

r

(

m−1, n

)

−

ψ

_k

( )

r

(

m−1, n

)

= Φ_k

( )

r

(

m−2, n

)

−_

ψ

( )

_kr

(

m−2, n

)

+

ψ

_k

( )

r

(

m−1, n

)

_

 

⋮ ⋮

( )

₍

₎

( )

(

)

gcd ,

,

1

r

r r

n m

r r _d

r k

d m n

m n d

k

ψ

_{= ∑}

µ

 − 

 ₋ 

592

( )

( )

( )

( )

1

1

1, m ,

r r

k k

i

n −

ψ

i n

=

= Φ − ∑

( )

( )

( )

1

1 _{gcd ,}

. 1

r r

r r

r

n n i

m

r _d r _d

r r

i

d n d i n

d d

k k

µ

−

µ

−

=

 

  _  _

   

= ∑ − ∑ _ ∑ _

  _  ₋ _

  _  _

which proves (14). 4. Conclusion

We count the exact number of nonempty subsets of { m , m+1 ,…, n} which are (i) r-relatively prime

(ii) r-relatively prime of cardinality k. (iii) r-relatively prime to n

(iv) r-relatively prime to n of cardinality k.

by using r-Generalized version of Mobius inversion formula for two variables.

Acknowledgements. The authors are grateful to the referee for valuable comments and interesting suggestions.

REFERENCES

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2. M.B.Nathanson, Affine invariants, relatively prime sets, and a phi function for subsets of {1, 2 , … , n}, Integer, 7 (2007) #A1,7pp. (electronic).

3. M.El.Bachraoui, The Number of relatively prime subsets and phi functions for {m, m+1, … n}, Integer, 7 (2007), #A43, 8pp. (electronic).

4. M.B.Nathanson, Elementary Methods in Number Theory, Graduate Texts in Mathematics, Vol. 195. Springer- New York, 2000.

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