Vol. 14, No. 3, 2017, 497-508
ISSN: 2279-087X (P), 2279-0888(online) Published on 13 November 2017
www.researchmathsci.org
DOI: http://dx.doi.org/10.22457/apam.v14n3a17
497
Annals of
r-Relatively Prime Sets and r-Generalization of Phi
Functions for Subsets of {1,2,…,n}
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: [email protected], [email protected]
1
Corresponding author.
Received 24 October 2017; accepted 9 November 2017
Abstract. A non-empty subset Aof positive integers {1,2,…,n} is said to be relatively prime if gcd
( )
A =1. Let r be a positive integer ≥1. A nonempty subset{
1, 2,...,}
A⊆ n is r-relatively prime if greatest rth power common divisor of elements of A is 1. In this case we write gcdr
( )
A =1. Note that gcd( )
A =1 implies( )
gcdr A =1 but the converse need not be true. Let f( )r
( )
n denotes the number of r-relatively prime subsets of{
1, 2, ..., n}
and fk( )r( )
n denotes the number of r-relatively prime subsets of{
1, 2, ..., n}
of Cardinality k. Φ( )
n denotes the number of non empty subsets A of{
1, 2, ..., n}
such that gcd A( )
is relatively prime to n. Φk( )
n denotes the number of non-empty subsets A of Cardinality k of {1,2,…,n} such that gcd A( )
is relatively prime to n. We define Φ( )r( )
n to be the number of non-empty subsets A of{
1, 2, ..., n}
such that greatest rth power common divisor of elements of A and n is 1.( )r
( )
k n
Φ is defined as the number of subsets A of
{
1, 2, ..., n}
such that Card A( )
=kand gcdr
( )
A is r-relatively to n. Φ( )r( )
n and Φ( )kr( )
n are r-generalizations of Φ( )
nand Φk
( )
n defined by Nathanson [2]. Exact formulas and asymptotic estimates are obtained for these functions. These results are extensions of results of Nathanson [2]. Some of our proofs use the r-Generalization of Mobius inversion formula.498 1. Introduction
For a nonempty subset Aof {1,2,…,n } let gcd A
( )
denote the gcdof the elements ofA. Nathanson [2] defined a non empty subset Aof {1,2,…,n} is relatively prime if
( )
gcd A =1.Let f n
( )
denote the number of relatively prime subsets of {1,2,…,n} and for k≥1, fk( )
n denote the number of relatively prime subsets of {1,2,…,n} of cardinality k. LetΦ( )
n denote the number of non empty subsets Aof {1,2,…,n} such that gcd A( )
is relatively prime to n and for integer k≥1,Φk( )
n denote the number of non empty subsets Aof {1,2,…,n} such that gcd A( )
is relatively prime to n and card (A)=k. Nathanson[2] obtained the exact formulas and Asymptotic estimates for these four fuctions. In this paper ,we define the functions f( )r( )
n , fk( )r( )
n ,Φ( )r( )
nand Φk( )r
( )
n and obtain exact formulas and Asympotic estimates for these four functions. We derive Mobius Inversion Formula r- Generalized Version to obtain Exact formulas.Definition 1.
Mobius function
r
-analogue( ) ( )
1 2 1 21 if 1
1 if ... where , , ..., are distinct primes 0 otherwise.
s r r r
r s s
n
n n p p p p p p
µ
=
= − =
Theorem 1. For all positive integersn≥2r,
( )
( )
2 2 2 . nr
r n
f n
≤ − (1)
For positive integers n≥2r and k,
( )
( )
2 .r k
n
r n
f n
k
k
≤ −
(2)
Proof: The set
{
1, 2, 3,..., 2 , ..., r n}
contains the set{
}
2 2 , 2 2 , ..., r × r nr2r which
has no subset that is
r
-relatively prime. Therefore among the 2n −1 non-empty subsets of{
1, 2, .., n}
those which contain any one of the 2[ / 2 ]n r −1 non-empty subsets of{
2 , 2 2 , ..., 2r 2}
r × r n r
499
( )
( )
2( )
1 2 2 1 2 2 2n n
r r
r n n
f n
≤ − − − = −
which proves (1). Similarly,
( )
( )
2 .
r k
n
r n
f n
k
k
≤ −
We now find a lower bound for f
( )
r( )
n and fk( )
r( )
n .If 1∈A A, ⊆
{
1, 2, .., n}
then A isr
-relatively prime. There are 2n−1 sets{
1, 2, ..,}
A⊆ n with 1∈A.Hence f
( )
r( )
n ≥ 2n−1. (3) Let n≥3 .r If 1∉A, 2r∈A, 3r∈A then A isr
-relatively prime and hencef
( )
r( )
n ≥ 2n−1+2n−3. (4) Let n≥5 .r If 1∉A, 3r∈A, but 2r∈A, 5r∈A then A isr
-relatively prime andthere are 2n−4 such subsets. Again 1∉A, 2r∈A, but 3r∈A, 5r∈A then A is
r
-relatively prime and there are 2n−4 such subsets. Hencef
( )
r( )
n ≥ 2n−1+2n−3+ ×2 2n−4 = 2n−1+2n−2. (5) Similarly
( )
( )
1 3 2 4 .1 2 2
k
r n n n
f n
k k k
− − −
≤ − + − + −
(6)
Exact formulas and asymptotic estimates
Let
[ ]
x denotes the greatest integer less than or equals to x. If x≥1 and n=[ ]
x then[ ]
xx n
d d d
= =
for all positive integers d.
2. Mobius inversion formula r- generalized version
Theorem 2. Let F
( )
r( )
x be a function defined for x≥1 and define the function( )
r( )
G x for x≥1 as
( )
( )
( )
1 r r
r r x
d d x
G x F
≤ ≤
= ∑
where the summation is over all
positive integers d where dr ≤x
and F
( )
r( )
x = =0 G( )
r( )
x if x∈( )
0, 1 . Then for all inters d whered
r≤
x
,
( )
( )
( )
( )
( )
( )
( )
1 1
r r
r r
r r x r r r x
r
d d
d x d x
G x F F x
µ
d G≤ ≤ ≤ ≤
= ∑ ⇔ = ∑
500 Proof: Assume
( )
( )
( )
1
. r r
r r x
d d x
G x F
≤ ≤ = ∑ Consider
( )
( )
( )
( )
( )
( )
1 1 1
r x
d
r r r
r r r
r r r x r r x
r r
d t d
d x d x t
G x
µ
d Gµ
d F≤ ≤ ≤ ≤ ≤ ≤ = ∑ = ∑ ∑
( )
( )
1 r r r r
r x r
r u
u x d u
F
µ
d≤ ≤ = ∑ ∑
( )
r( )
.F x
=
Conversely, Assume
( )
( )
( )
( )
1
. r r
r r r x
r
d d x
F x
µ
d G≤ ≤
= ∑
Consider
( )
( )
( )
( )
1 1 1
r x d
r r r
r r r
r x r r r x
r r
d t d
d x d x t
F
µ
dµ
d G≤ ≤ ≤ ≤ ≤ ≤ = ∑ ∑ ∑
( )
( )
1 r r r r
r x r
r u
u x t u
G
µ
t≤ ≤ = ∑ ∑
=G
( )
r( )
x . Theorem 3. For all positive integers n≥2r ,(i)
( )
12 1
r r
r n n
d d n f ≤ ≤ = − ∑
(7)
and (ii)
( )
( )
( )
12 1 .
n r d r r r r d n
f n
µ
d ≤ ≤ = ∑ − (8)
For all positive integers n≥2r, k and
r
≥
1
, (i)( )
r1 r r x k d d n n f k ≤ ≤ = ∑
(9)
and (ii)
( )
( )
( )
1 r r n r r d r k d n
f n d
501
Proof: Let A be a non-empty subset of
{
1, 2, .., n}
and greatest rth power commondivisor of A is dr.Then A1 1r *A ar a A d
d
= = ∈
is
r
-relatively prime subset of{
1, 2, .., nr}
.d
Conversely, if 1
A is
r
-relatively prime subset of{
1, 2, .., nr}
d ,then A=dr*A1=
{
dr⋅a a∈A1}
is a non-empty subset of{
1, 2, .., n}
with greatest thr power common divisor of A equals todr.Therefore it follows that there are exactly
( )
r r n
d
f
subsets of
{
1, 2, .., n}
with greatest thr power common divisordr and
hence
( )
1
2 1
r r
r n n
d d n
f ≤ ≤
= −
∑
which proves (7). We apply Theorem (2) to the
functionF( )r
( )
x = f( )r( )
[ ]
x for all x≥1we define( )
( )
( )
1 r r
r r x
d d x
G x F
≤ ≤
= ∑
( )
1 r r
r x
d d x
f ≤ ≤
= ∑
[ ]
2 x 1.
= −
By Theorem (2)
( )
( )
[ ]
( )
( )
( )
( )
1
r r
r r r r x
r
d d x
f x F x
µ
d G≤ ≤
= = ∑
( )
12 1
x r d r
r r d x
d
µ
≤ ≤
= ∑ −
( )
( )
( )
12 1
n r d r
r r
r d n
f n n
µ
d
≤ ≤
= ∑ −
which proves (8).
We now prove (9) and (10).
Note thatfk
( )
r( )
n =#{
A⊆{
1, 2, ..., n}
: Card A=k, gcdr( )
A =1 .}
Let A⊆
{
1, 2, ..., n}
withCard A= k and greatest rth power common divisor of A is equals to dr. Let A1 1 *r A d
= =
{
ar}
. d a∈A Then1
1, 2, ..., r n d
A ⊆
502
CardA1=Card A=kand
gcd
r( )
A
1=
1.
Conversely,if 1 1, 2, ..., r n dA ⊆
and
( )
1 1
Card A =k,gcdr A =1thenA=dr*A1is such thatCard A=kand
( )
gcdr A =dr.
There is
1 1
−
correspondence betweenr
-relatively prime subsets 11, 2, ..., nr d
A ⊆
of Cardinality k and the non-empty subsets A of
{
1, 2, ..., n}
with gcdr
( )
A =dr and Card A=k. Hence( )
r 1r r
n k
d d n
n f
k ≤ ≤
= ∑
which
proves (9).
By Theorem (1) we have
( )
( )
( )
1
r r
n
r r
d r
k
d n
f n d
k
µ
≤ ≤
= ∑
which proves (10).
Theorem 4. For all positive integers n≥2r , r we have
2 2 2 2 3
( )
( )
2 2 2 .n
n n
r
r r r
n n
n f n
− − ⋅ ≤ ≤ − (11)
Proof: For n≥2r we have by equation (7)
( )
1
2 1 r
r r
n n
d d n
f
≤ ≤
− = ∑
This implies
( )
( )
[ ]
( )
( )
2 3
2 1
r r r
r r r
n n n
d d n
f n f f
≤ ≤
= + + ∑ +
( )
( )
2 2 2 3 n nr r
r
f n n
≤ + + ⋅
combining this with equality (1),2 2 2 2 3
( )
( )
2 2 2 nn n
r
r r r
n n
n f n
− − ⋅ ≤ ≤ − .
Theorem 5. For all positive integers n≥2r, k and r
2 3
( )
( )
2 .
r r r
k
n n n
r
n n
n f n
k k
k k k
− − ≤ ≤ −
(12)
Proof: By equation (9)
( )
1r r
r n
k d d n
n f
k ≤ ≤
= ∑
503
Therefore
( )
( )
[ ]
( )
( )
2r 3r r r
r r n r n
k k k
d d n
n
f n f f
k ≤ ≤
= + +
∑
( )
( )
2 3 .
r r
n n
r k
f n n
k k
≤ + +
Therefore 2 3
( )
( )
2
r r r
k
n n n
r
n n
n f n
k k
k k k
− − ≤ ≤ −
by equation (2).
3. A Phi function for sets and its r-generalization
Nathanson [2] defined Phi function for sets, denoted by Φ
( )
n to be the number of non-empty subsets A of{
1, 2, 3, ..., n}
such thatgcd A
( )
is relatively prime to n. For example for distinct primes p and q we have
( )
( )
( )
2
2
2p 2, 2p 2 , p 2pq 2p 2q 2.
p p pq
Φ = − Φ = − Φ = − − +
Note that Φ1
( )
n =ϕ
( )
n for all n≥1.We define, for a positive integer r≥1,Φ( )
r( )
nto be the number of non-empty subsets A of
{
1, 2, ..., n such that greatest}
rth power common divisor of A and n is 1. For example for distinct primes p and q,Φ
( )
r( )
pr =2pr −2
( )
( )
2 22 r 2 r
r r p p
p
Φ = −
Φ
( )
r( )
p qr r =2p qr r −2qr −2pr +2.Corollary 6. If F
( )
r( )
n and G( )
r( )
n are arithmetic functions, then( )
( )
( )
( )
( )
( )
( )
.r r
r r
r
r r n r r n
r
d d
d n d n
G n = ∑ F ⇔ F n = ∑
µ
d G
Proof: Assume
( )
( )
( )
r . rr r n
d d n
G n = ∑ F
Consider
( )
( )
r r r
r r r n
r d
r r
n n
r r
d t d
d n d n t
d F
µ
µ
=
∑ ∑ ∑
504
( )
( )
( )
( )
.r
r r r
r n r r
r u
u n d u
F
µ
d F n= ∑ ∑ =
Conversely, assume
( )
( )
( )
( )
.r r
r r r n
r
d d n
F n = ∑
µ
d G Consider
( )
( )
( )
r r r
r r r n
r d
r n r r n
r
d t d
d n d n t
F
µ
t G
=
∑ ∑ ∑
( )
( )
( )
( )
.r
r r r
r n r r
r u
u n t u
G
µ
t G n
= ∑ ∑ =
Theorem 7. For all positive integers n, r≥1,
( )
r 2 1.r
r n n
d d n
Φ = −
∑
(13)
Also Φ
( )
r( )
1 =1 and for n≥2r,( )
( )
( )
2 1 . nr d r
r r
r d n
n
µ
d
Φ = ∑ −
(14)
Proof: For every rth power divisor dr of n we define the function ψ
( )
r(
n d,)
to be the number of non-empty subsets A of{
1, 2, ..., n}
such that greatest rth power common divisor of A and n is dr, i.e.( )
r(
,)
#{
{
1, 2, ...,}
: , and gcd(
{ }
)
r}
. rn d A n A A n d
ψ
= ⊆ ≠φ
∪ =Then
( )
r(
,)
( )
r rn n d
d
ψ
= Φ and hence
( )
(
)
( )
2 1 , .
r
r r
r r
n n
d
d n d n
n d
ψ
− = ∑ = ∑ Φ
which proves (13). This implies
( )
( )
( )
2 1n r d r
r r
r d n
n
µ
d
Φ = ∑ −
(by using corollary
(6)) which proves (14).
Theorem 8. For positive integersn≥2r, k and r
( )
kr r
r n
d d n
n k
Φ =
∑
505
and
( )
k( )
( )
r . rn
r r d
r d n
n d
k
µ
Φ = ∑
(16)
Proof: For every rth power divisor dr of n we define
ψ
k( )
r(
n d,)
to be number of subsets A of{
1, 2, ..., n}
of Cardinality k such that greatest rth power commondivisor of A and n is dr. That is
k
( )
r(
,)
#{
{
1, 2, ...,}
: # and gcd(
r( ) { }
)
r}
. rn d A n A k A n d
ψ
= ⊆ = ∪ =Note that k
( )
r(
,)
( )
kr rn n d
d
ψ
= Φ
( )
(
,)
( )
r
r r
r r n
k k
d
d n d n
n n d
k
ψ
= = Φ ∑ ∑
which proves (15).
Using Corollary (6), we get
( )
k( )
( )
r rn
r r d
r d n
n d
k
µ
Φ = ∑
which proves (16).
Theorem 9. Letn≥2r. If n is odd then
( )
( )
2 .2 3 . nr
r n
n n
Φ = + Ο
(17)
If 2r n ,
( )
( )
2 22 .2 3 . n nr r
r n
n n
Φ = − + Ο
(18)
Proof: We have
( )
( )
{
{
}
( )
}
( )
1
gcd , 1.
# 1, 2, ..., , gcd r
r r
n
r r
r d
d n
n A n A
φ
A d= =
Φ = ∑ ⊆ ≠ = where
the summation is over rth power of integer d which satisfy the condition1≤dr ≤n
( )
( )
1
gcd , 1.
r r
r r
n r
n d d
d n
f =
=
= ∑
506
If n is odd, then
( )
( )
( )
( )
( )
( )
( )
2 3
gcd , 1
r r r r
r r
n
r r r n r n
d d n
d n
n f n f f
≤ ≤ =
Φ = + + ∑
3 3
2 2 4
2 2 .2 2 2 .2
n n
n n n
r r
r r r
n
n n
= − + Ο + + Ο + Ο
3
2 .2
n r n
n
= + Ο
(Using equation (11)) which proves (17).
If 2r n, then
( )
( )
( )
( )
( )
( )
( )
2 3
gcd , 1
r r r r
r r
n
r r r n r n
d d n
d n
n f n f f
≤ ≤ =
Φ = + + ∑
2 22 .2 3 .2 3
n n
n
r r
r n
n n
= − + Ο + Ο
2 22 .2 3 n n
r r
n
n
= − + Ο
which proves (18)
Theorem 10. If 2r n and n is sufficiently large then
( )
( ) 3 .r k
n
r n
n n
k
k
Φ = + Ο
(19)
If2r n then
( )
( ) 2 3 .
r r
k
n n
r n
n n
k k
k
Φ = − + Ο
(20)
Proof: We have
( )
( )
{
{
}
( )
}
( )
1
gcd , 1
# 1, 2, ..., : # , gcd r
r r
r r
r k
d n d n
n A n A k A d
≤ ≤ =
507
( )
( )
1
gcd , 1
. r r
r r
r n
k d d n
d n
f
≤ ≤ =
= ∑
By theorem (4),
( )
( )
2 3
r r
k
n n
r n
f n n
k
k k
= − + Ο
( )
( )
23 Therefore
r r
k
n n
r n
n n
k k
k
Φ = − + Ο
2 4 6 3
r r r r
n n n n
n n
k k k k
+ − + Ο + Ο
2 3 if 2 .
r r
n n
r
n
n n
k k
k
= − + Ο
If 2r n
( )
( )
3 .r k
n
r n
n n
k
k
Φ = + Ο
4. Conclusion
In conclusion we have obtained effective formulas to get the exact number of r-relatively prime subsets of {1,2,…,n} and number of non empty subsets A of {1,2,…,n} such that A is r-relatively prime to n by deriving Mobius inversion formula r- generalised version.
Acknowledgement. We are grateful to the referees for suggesting some corrections and valuable comments.
REFERENCES
1. M.B.Nathanson, Elementary Methods in Number Theory, Graduate Texts in Mathematics, Vol. 195. Springer-Verlag, New York, (2000).
2. Melvyn B.Nathanson, Affine invariants, relatively prime sets, and a Phi function for subsets of {1,2,…,n}, Integers, 7(2007), A1, 7pp.
3. V.Kavitha and R.Govindarajan, A study on Ramsey numbers and its bounds, Annals of Pure and Applied Mathematics, 8(2) (2014) 227-236.
508
5. M.El.Bachraoui, On relatively prime subsets, combinatorial identities, and diaphontine equations, J. Integers Seq., 15 (2012) Article 12.3.6, 9 pp.
