INTEGERS 10 (2010), 465-476
#A39
ON RELATIVELY PRIME SETS COUNTING FUNCTIONS
Temba Shonhiwa
The John Knopfmacher Centre for Applicable Analysis and Number Theory,
School of Mathematics, University of the Witwatersrand, South Africa
Received: 5/11/09, Revised: 4/14/10, Accepted: 4/26/10, Published: 9/15/10
Abstract
This work is motivated by Nathanson’s recent paper on relatively prime sets and a
phi function for subsets of {1, 2, 3, . . . , n}. We establish enumeration formulas for
the number of relatively prime subsets and the number of relatively prime subsets
of cardinality k of {1, 2, 3, . . . , n} under various constraints. Further, we show how
this work links up with the study of multicompositions.
1. Background
Our paper is motivated by a recent paper of Nathanson [8] who defined a nonempty
subset A of {1, 2, . . . , n} to be relatively prime if gcd(A) = 1. He defined f (n) to be
the number of relatively prime subsets of {1, 2, . . . , n} and, for k ≥ 1, fk (n) to be
the number of relatively prime subsets of {1, 2, . . . , n} of cardinality k. Further, he
defined Φ(n) to be the number of nonempty subsets A of the set {1, 2, . . . , n} such
that gcd(A) is relatively prime to n and, for integer k ≥ 1, Φk (n) to be the number
of subsets A of the set {1, 2, . . . , n} such that gcd(A) is relatively prime to n and
card(A) = k. He obtained explicit formulas for these functions and deduced asymptotic estimates. These functions were subsequently generalized by El Bachraoui [5]
to subsets A ∈ {m + 1, m + 2, . . . , n} where m is any nonnegative integer, and then
by Ayad and Kihel [3] to subsets of the set {a, a + b, . . . , a + (n − 1)b} where a and
b are any integers.
El Bachraoui [4] defined for any given positive integers l ≤ m ≤ n, Φ([l, m], n) to
be the number of nonempty subsets of {l, l + 1, . . . , m} which are relatively prime
to n and Φk (l, m], n) to be the number of such subsets of cardinality k. He found
formulas for these functions when l = 1 [4].
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2. Introduction
It turns out that some of Nathanson’s results are special cases of number theoretic
functions investigated by Shonhiwa. In [10], Shonhiwa defined and investigated the
following functions and established the following result.
Theorem 1 Let
Skm (n) =
!
1; ∀ n ≥ k ≥ 1, m ≥ 1
(1)
!
1; ∀ n ≥ k ≥ 1,
(2)
1≤a1 ,a2 ,...,ak ≤n
(a1 ,a2 ,...,ak ,m)=1
Gk (n) =
1≤a1 ,a2 ,...,ak ≤n
(a1 ,a2 ,...,ak )=1
!
Lm
k (n) =
1≤a1 ≤a2 ≤···≤ak ≤n
(11 ,a2 ,...,ak ,m)=1
1; ∀ n ≥ k ≥ 1, m ≥ 1
(3)
and
!
Tkm (n) =
1≤a1 <a2 <···<ak ≤n
(a1 ,a2 ,...,ak ,m)=1
1; ∀ n ≥ k ≥ 1, m ≥ 1.
(4)
Then
Skm (n) =
!
µ(d)
d|m
Lm
k (n)
=
!
µ(d)L1k
d|m
$" n #%
d
=
" n #k
!
d|m
d
,
&' n (
)
+k−1
d
µ(d)
,
k
and
Tkm (n) =
!
d|m
µ(d)Tk1
$" n #%
d
=
!
d|m
&' n ()
µ(d) d .
k
From above, it follows that
Φk (n) = Tkn =
! $ n % &d)
µ
d
k
d|m
(5)
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and
Φ(n) =
n
!
Tkn (n) =
k=1
!
n
(6)
µ(d)2 d ,
d|m
as shown therein and as proved in [8].
3. Main Results
The result obtained concerning the function Gk (n) in [10] is incorrect and we provide
the correction below. The corrected result makes use of the following theorem [1].
Theorem 2 (Generalized Möbius inversion formula) If α is completely multiplicative we have
$x%
$x%
!
!
G(x) =
α(n)F
⇐⇒ F (x) =
µ(n)α(n)G
.
n
n
n≤x
n≤x
We may now prove our first result as follows.
Theorem 3 We have
Gk (n) =
!
µ(j)
j≤n
Proof.
Since
Gk (n) = nk −
we have
* +k
n
.
j
!
!
1 = nk −
n
!
= nk −
n
!
j=2 1≤a1 ,a2 ,...,ak ≤n
(a1 ,a2 ,...,ak )=j
n
!
j=1
Gk
$" n #%
d
= nk .
!
1
j=2 1≤b1 ,b2 ,...,bk ≤( n )
j
(b1 ,b2 ,...,bk )=1
j=2
Gk
&* +)
n
,
j
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Hence, by Theorem 2, it follows that
* +k
j * +k
n
!
!
n
n
G(n) =
Gk (n) =
µ(j)
=
µ(j)
j
j
j=1
k=1
k=1 j=1
k=1
" #&
" #j )
n
n
µ(j)
1
−
n
j
j
!
$
" #%
=
.
1 − nj
j=1
n
!
n !
n
!
!
Using our definition, results from [10], Nathanson’s notation and arguing as
above, it also follows that
fk (n) =
!
1≤a1 <a2 <···<ak ≤n
(a1 ,a2 ,...,ak )=1
& ) !
&* +)
n
n
n
1=
−
fk
,
k
j
j=2
which gives
n
!
fk
j=1
&* +) & )
n
n
=
.
j
k
Thus,
" #
& n )
fk (n) =
µ(j) j .
k
j=1
n
!
From this it follows that
" #
& n )
f (n) =
fk (n) =
µ(j) j
k
k=1
k=1 j=1
" #
j & n )
n
n
$ n
%
!
!
!
j
=
µ(j)
=
µ(j) 2( j ) − 1 .
k
j=1
j=1
k=1
n
!
n !
n
!
We now prove our next theorem.
Theorem 4 Let
Hk (n) =
!
1≤a1 ≤a2 ≤···≤ak ≤n
(a1 ,a2 ,...,ak )=1
1.
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Then
" #
& n
)
j +k−1
Hk (n) =
µ(j)
.
k
j≤n
!
Proof.
Arguing as above it follows that
&
) !
&* +)
n
n+k−1
n
Hk (n) =
−
Hk
k
j
j=2
which implies
n
!
j=1
Hk
&* +) &
)
n
n+k−1
=
,
j
k
and hence, by Theorem 2, we obtain the result
" #
)
j & n
n
n
!
!
!
j +k−1
H(n) =
Hk (n) =
µ(j)
k
j=1
k=1
k=1
"
#
& n
) !
n
n
!
+j
=
µ(j) j
−
µ(j).
j
j=1
j=1
!
We now define the corresponding totient function as
! &n)&d + k − 1)
Ψk (n) = Lnk (n) =
µ
.
d
k
d|n
Then
Ψ(n) =
n
!
Ψk (n) =
k=1
n
!
k=1
Lnk (n) =
)
n &
! &n) !
d+j−1
d|n
d
k=1
j
=
! &n)&d + n)
,
d
n
d|n
or equivalently,
& ) !
)
∞ &
∞
!
2n
2n n !
xn
1
=
Ψ(d) ⇐⇒
x =
Ψ(n)
=√
− 1.
n
n
n
1−x
1 − 4x
n=1
n=1
d|n
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It turns out the function Tkm (n) relates to other functions connected with the
study of compositions of n into relatively prime summands as follows.
Gould [6] investigated the function
& ) ! $ %&
)
!
!
n
n
d−1
Rk (n) =
1=
Ck (d)µ
=
µ
,
d
d
k−1
1≤a1 +a2 +···+ak =n
(a1 ,a2 ,...,ak )=1
d|n
d|n
,
where Ck (n) = n−1
k−1 and obtained many other significant results concerning this
function. Consequently,
!
Tkn (n) =
1
1≤a1 <a2 <···<ak ≤n
(a1 ,a2 ,...,ak ,n)=1
! $ n % &d)
=
µ
d
k
d|n
.&
) &
)/
! $n%
d−1
d−1
=
µ
+
d
k−1
k
d|n
= Rk (n) + Rk+1 (n).
Therefore, we may obtain results concerning either function by using known properties of the other. In particular, we may obtain the Lambert series for Tkn (n) as
follows:
∞
!
Tkn
n=1
xn
1 − xn
=
∞
!
Rkn
n=1
=
=
=
∞
!
xn
xn
n
+
R
1 − xn n=1 k+1 1 − xn
xk
(1 − x)k+1
∞
∞
!
!
xn
Ck (n)xn
n=0
∞
!
xn
n=1
n=0
n
!
j=0
Ck (n − j),
which is equivalent to
!
d|n
as expected.
Tkd (d) =
n
!
j=0
Ck (n − j) =
& )
n
,
k
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The inverse function of Rk (n) is
& )* +
n
!
n
j
(−1)n−j
j
k
Ak (n) =
(7)
j=k
and it is shown in [6] that these two satisfy the orthogonality relations.
Theorem 5 We have
n
!
Rk (j)Aj (n) = δkn
j=k
and
n
!
Rj (n)Ak (j) = δkn .
j=k
In [10], it is shown that the inverse of Tkn (n) is
&
)* +
n
!
j
n−j n + 1
Kk (n) =
(−1)
,
j+1
k
j=k
and that:
Theorem 6 We have
n
!
Tkj (j)Kj (n) = δkn and
j=k
n
!
Kk (j)Tjj (n) = δkn .
j=k
It follows that
.&
) & )/
n
!
n
n
(−1)n−j
+
j+1
j
j=k
&
)
n
!
n
=
(−1)n−j
+ Ak (n),
j+1
Kk (n) =
j=k
so that
Ak (n) = Kk (n) + Kk (n − 1).
(8)
Whilst we have a closed form expression for Ak (n) it does not reveal enough regarding the structure of Ak (n). Our next result responds to this concern for a special
case of Ak (n).
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Theorem 7 For n ≥ 1, we have
&
)
n+k−1
Ak (k + n) = (−1)
; ∀ k ≥ n + 1.
k−1
n
Proof.
From Tkn (n) = Rk (n) + Rk+1 (n) and Theorem 3.4 above, it follows that
n
!
n
Aj (n)Tkj (j) = δkn + δk+1
.
(9)
j=k
So that for n = k,
Ak (k)Tkk (k) = 1 implies Ak (k) = Tkk = 1; ∀ k ≥ 1.
And for n = k + 1,
k+1
Ak (k + 1)Tkk (k) + Ak+1 (k + 1)Tk+1
(k + 1) = 1 =⇒ Ak (k + 1) = −k; ∀ k ≥ 1.
For n ≥ k + 2 we may rewrite equation (9) above as
Ak (n) +
Tkn (n)
=−
n−1
!
Aj (n)Tkj (j),
j=k+1
then for n = k + 2,
Ak (k + 2) + Tkk+2 (k + 2) = −Ak+1 (k + 2)Tkk+1 (k + 1)
= (k + 1)Tkk+1 (k + 1)
= (k + 1)2 ,
since Ak (k + 1) = −k. Hence
Ak (k + 2) = (k + 1) −
2
= (k + 1)2 −
=
!
d|k+2
& k+2 )
µ(d) d
k
(k + 1)(k + 2)
2
(−1)2 k(k + 1)
2
provided k ≥ 3.
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Now assume the result holds for k + 1, k + 2, . . . , k + n − 1 and consider
Ak (k + n) + Tkk+n (k + n) = −
=
n+k−1
!
Aj (n + k)Tkj (j)
j=k+1
n−1
!
(−1)n−i Tkk+i (k + i)
(n − i)!
i=1
=
n−1
1
(k + j)
j=1
n−1
0
(k + j)
j=i
2n−1
3
! (−1)n+1−i (k + i)
i=1
(n − i)!i!
.
Then
Ak (k + n) =
n−1
0
j=1


(k + j)  n
& )

!
n
(−1)n−j−1
(k + j)


n!
j
= (−1)n
j=1
&
)
n+k−1
,
k−1
where we have used the inductive hypothesis as well as assumed that k ≥ n + 1. !
We note in passing that
Rk (k + n) = (−1)n Ak (k + n) for all k ≥ n + 1.
Following up on Gould’s paper, Andrews [2] introduced the function gm (n), which
gives the number of m-compositions of n with relatively prime positive summands
so that
n
!
T (n) =
Rk (n) = g1 (n).
k=1
It is shown in [2] that the total number of m-compositions of n is (m + 1)n−1 and
hence,
!
(m + 1)n−1 =
gm (d).
d|n
In a follow-up paper, Shonhiwa [11] provided an alternative investigation of the
function gm (n).
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From the equation
! $ n % &d)
µ
d
j
Tjn =
d|n
it follows that
n
!
Tjn (n)xj =
j=1
! $n%
d
d|n
(x + 1)d for all n ≥ 2.
Therefore
! &n)
gm (n) =
µ
(m + 1)d−1
d
d|n
= (m + 1)−1
! &n)
µ
(m + 1)d
d
d|n
= (m + 1)
−1
n
!
j=1
Tjn (n)M j for all n ≥ 1.
In particular, for m = 1, we obtain
! &n)
g1 (n) = T (n) =
µ
2d−1
d
d|n
! &n)
=
µ
(3 − 1)d−1
d
d|n
=
)
d−1 &
! &n) !
d−1 j
µ
3 (−1)d−1−j
d j=0
j
d|n
≡ 0 (mod 3) for all n ≥ 3
(see [7]).
Further, from
gm (i) = (m + 1)−1
i
!
Tji (i)mj ,
j=1
it follows that
Ki (n)gm (i) = (m + 1)−1
i
!
j=1
Tji (i)Ki (n)mj ;
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which implies that
n
!
n !
i
!
Ki (n)gm (i) = (m + 1)−1
i=1
Tji (i)Ki (n)mj
i=1 j=1
= (m + 1)−1
n
!
mj δjn
j=1
= (m + 1)
−1
mn ,
from above.
Hence
n−1
gm (n) =
!
mn
−
Ki (n)gm (i)
m + 1 i=1
n−1
=
n−1
!
!
mn
+
Ki (n − 1)gm (i) −
Ai (n)gm (i)
m + 1 i=1
i=1
n−1
=
mn
mn−1 !
+
−
Ai (n)gm (i)
m + 1 m + 1 i=1
= mn−1 −
as expected; see [11].
n−1
!
Ai (n)gm (i),
i=1
Acknowledgment Unfortunately Dr. T. Shonhiwa passed on during the review
process of this manuscript. The following is a statement from his widow:
This publication is dedicated to the memory of Temba Shonhiwa’s love of
Mathematics and the development of a Mathematical Sciences Community in Africa. May his soul rest in peace. We would like to acknowledge
and thank Dr. Augustine Munagi for finalizing and resubmitting this
script.
Many thanks,
Fortune Shonhiwa
References
[1] T. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York, LLC, 1995.
[2] G. Andrews, The theory of compositions, IV: Multicompositions. The Mathematics Student(Special Centenary Volume):25-31 (2007).
INTEGERS: 10 (2010)
476
[3] M. Ayad and O. Kihel, On the number of subsets relatively prime to an integer, J. Integer
Seq., 11 (2008) Article 08.5.5.
[4] M. El Bachraoui, On the number of subsets of [1, m] relative prime to n and asymptotic
estimates, Integers, 8 (2008), A41, 5 pp. (electronic).
[5] M. El Bachraoui, The number of relatively prime subsets and phi functions for {m, m +
1, . . . , n}, Integers, 7 (2007), A43, 8 pp. (electronic).
[6] H.W. Gould, Binomial Coefficients, the Bracket Function, and Compositions with Relatively
Prime Summands, The Fibonacci Quarterly, Volume 2, Number 4, 1964.
[7] H.W. Gould, Remarks on Compositions of Numbers into Relatively Prime Parts, Notes on
Number Theory and Discrete Mathematics, 11 (2005), No. 3, 1–6.
[8] M.B. Nathanson, Affine invariants, relative prime sets, and a phi function for subsets of
{1, 2, . . . , n}, Integers, 7 (2007), A1, 7 pp. (electronic).
[9] M.B. Nathanson and B. Orosz, Asymptotic estimates for phi functions for subsets of {M +
1, M + 2, . . . , N }, Integers, 7 (2007), A54, 5 pp. (electronic).
[10] T. Shonhiwa, A Generalization of the Euler and Jordan Totient Functions, The Fibonacci
Quarterly, Volume 37, February 1999, Number 1.
[11] T. Shonhiwa, On Compositeness in Multicompositions, Quaestiones Mathematicae, 32
(2009), 1–11.