Journal of Inequalities in Pure and
Applied Mathematics
A NOTE ON MULTIPLICATIVELY e-PERFECT NUMBERS
LE ANH VINH AND DANG PHUONG DUNG
School of Mathematics
University of New South Wales
Sydney 2052 NSW, Australia.
EMail: vinh@maths.unsw.edu.au
School of Finance and Banking
University of New South Wales
Sydney 2052 NSW, Australia.
EMail: phuogdug@yahoo.com
volume 7, issue 3, article 99,
2006.
Received 30 March, 2005;
accepted 03 February, 2006.
Communicated by: J. Sándor
Abstract
Contents
JJ
J
II
I
Home Page
Go Back
Close
c
2000
Victoria University
ISSN (electronic): 1443-5756
313-05
Quit
Abstract
Let Te (n) denote the product of all exponential divisors of n. An integer n is
called multiplicatively e-perfect if Te (n) = n2 and multiplicatively e-superperfect
if Te (Te (n)) = n2 . In this note, we give an alternative proof for characterization
of multiplicatively e-perfect and multiplicatively e-superperfect numbers.
A Note on Multiplicatively
e-perfect Numbers
2000 Mathematics Subject Classification: 11A25, 11A99.
Key words: Perfect number, Exponential divisor, Multiplicatively perfect, Sum of divisors, Number of divisors.
Le Anh Vinh and
Dang Phuong Dung
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Proof of Theorem 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
3
4
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 2 of 6
J. Ineq. Pure and Appl. Math. 7(3) Art. 99, 2006
http://jipam.vu.edu.au
1.
Introduction
Let σ(n) be the sum of all divisors of n. An integer n is called perfect if
σ(n) = 2n and superperfect if σ(σ(n)) = 2n. If n = pα1 1 · · · pαk k is the prime
factorization of n > 1, a divisor d | n, called an exponential divisor (e-divisor)
of n is d = p1β1 · · · pβkk with βi | αi for all 1 ≤ i ≤ k. Let Te (n) denote the product of all exponential divisors of n. The concepts of multiplicatively e-perfect
and multiplicatively e-superperfect numbers were first introduced by Sándor in
[1].
Definition 1.1. An integer n is called multiplicatively e-perfect if Te (n) = n
and multiplicatively e-superperfect if Te (Te (n)) = n2 .
2
A Note on Multiplicatively
e-perfect Numbers
Le Anh Vinh and
Dang Phuong Dung
In [1], Sándor completely characterizes multiplicatively e-perfect and multiplicatively e-superperfect numbers.
Title Page
Theorem 1.1 ([1]).
Contents
a) An integer n is multiplicatively e-perfect if and only if n = pα , where p is
prime and α is a perfect number.
b) An interger n is multiplicatively e-superperfect if and only if n = pα ,
where p is a prime, and α is a superperfect number.
JJ
J
II
I
Go Back
Close
Sándor’s proof is based on an explicit expression of Te (n). In this note, we
offer an alternative proof of Theorem 1.1.
Quit
Page 3 of 6
J. Ineq. Pure and Appl. Math. 7(3) Art. 99, 2006
http://jipam.vu.edu.au
2.
Proof of Theorem 1.1
a) Suppose that n is multiplicatively e-perfect; that is Te (n) = n2 . If n has
more than one prime factor then n = pα1 1 · · · pαk k for some k ≥ 2, αi ≥ 1 and
p1 , . . . , pk are k distinct primes. We have three separate cases.
1. Suppose that α1 = · · · = αk = 1. Then d is an exponential divisor of n if
and only if d = p1 · · · pk = n. Hence Te (n) = n, which is a contradiction.
2. Suppose that two of α1 , . . . , αk are greater 1. Without loss of generality, we may assume that α1 , α2 > 1. Then d1 = p1 pα2 2 · · · pαk k , d2 =
pα1 1 p2 pα3 3 · · · pαk k , d3 = n are three distinct exponential divisors of n. Hence
1 +1
d1 d2 d3 | Te (n). However, p2α
| d1 d2 d3 so Te (n) 6= n2 , which is a con1
tradiction.
3. Suppose that there is exactly one of α1 , . . . , αk which is greater than 1.
Without loss of generality, we may assume that α1 > 1 and α2 = · · · =
αk = 1. We have that if d is an exponential divisor of n then d =
pβ1 1 p2 · · · pk for some β1 | α1 . Hence if n has more than two distinct
2
1 2
exponential divisors then p32 | Te (n) = p2α
1 p2 · · · pk , which is a contradiction. However, d1 = p1 p2 · · · pk , d2 = pα1 1 p2 p3 · · · pk are two distinct
exponential divisors of n so d1 , d2 are all exponential divisors of n. Hence
2
1 2
Te (n) = pα1 1 +1 p22 · · · p2k = p2α
1 p2 · · · pk . This implies that α1 = 1, which
is a contradiction.
Thus n has only one prime factor; that is, n = pα for some prime p. In this
case then Te (n) = pσ(α) . Hence Te (n) = n2 = p2α if and only if σ(α) = 2α.
This concludes the proof.
A Note on Multiplicatively
e-perfect Numbers
Le Anh Vinh and
Dang Phuong Dung
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 4 of 6
J. Ineq. Pure and Appl. Math. 7(3) Art. 99, 2006
http://jipam.vu.edu.au
b) Suppose that n is multiplicatively e-superperfect; that is Te (Te (n)) = n2 . If
n has more than one prime factor then n = pα1 1 · · · pαk k for some k ≥ 2, αi ≥ 1
and p1 , . . . , pk are k distinct primes. We have two separate cases.
1. Suppose that α1 = · · · = αk = 1. Then d is an exponential divisor of n if
and only if d = p1 · · · pk = n. Hence Te (n) = n and Te (Te (n)) = Te (n) =
n which is a contradiction.
2. Suppose that there is at least one of α1 , . . . , αk which is greater 1. Without
loss of generality, we may assume that α1 > 1. Then d1 = p1 pα2 2 · · · pαk k ,
d2 = n = pα1 1 pα2 2 pα3 3 · · · pαk k , are two distinct exponential divisors of n.
2
k
Hence d1 d2 | Te (n). However, d1 d2 = pα1 1 +1 p2α
· · · p2α
so Te (n) =
2
k
γ1
γk
p1 · · · pk where γ1 ≥ α1 + 1, γi ≥ 2αi ≥ 2 for i = 2, . . . , k. Thus,
t1 = pγ11 p2 pγ33 · · · pγkk and t2 = Te (n) = pγ11 pγ22 pγ33 · · · pγkk are two distinct
exponential divisors of Te (n). Hence t1 t2 | Te (Te (n)). However, p12γ1 |
t1 t2 and γ1 > α1 , which is a contradiction.
Thus n has only one prime factor; that is n = pα for some prime p. In this
case then Te (n) = pσ(α) and Te (Te (n)) = pσ(σ(n)) . Hence Te (Te (n)) = n2 =
p2α if and only if σ(σ(α)) = 2α. This concludes the proof.
Remark 1. In an e-mail message, Professor Sándor has provided the authors
some more recent references related to the arithmetic function Te (n), as well
as connected notions on e-perfect numbers and generalizations. These are [2],
[3], and [4].
A Note on Multiplicatively
e-perfect Numbers
Le Anh Vinh and
Dang Phuong Dung
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 5 of 6
J. Ineq. Pure and Appl. Math. 7(3) Art. 99, 2006
http://jipam.vu.edu.au
References
[1] J. SÁNDOR, On multiplicatively e-perfect numbers, J. Inequal. Pure Appl.
Math., 5(4) (2004), Art. 114. [ONLINE: http://jipam.vu.edu.au/
article.php?sid=469]
[2] J. SÁNDOR, A note on exponential divisors and related arithmetic functions, Scientia Magna, 1 (2005), 97–101. [ONLINE http://www.
gallup.unm.edu/~smarandache/ScientiaMagna1.pdf]
[3] J. SÁNDOR, On exponentially harmonic numbers, to appear in Indian J.
Math.
[4] J. SÁNDOR AND M. BENCZE, On modified hyperperfect numbers,
RGMIA Research Report Collection, 8(2) (2005), Art. 5. [ONLINE: http:
//eureka.vu.edu.au/~rgmia/v8n2/mhpn.pdf]
A Note on Multiplicatively
e-perfect Numbers
Le Anh Vinh and
Dang Phuong Dung
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 6 of 6
J. Ineq. Pure and Appl. Math. 7(3) Art. 99, 2006
http://jipam.vu.edu.au