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volume 7, issue 3, article 99, 2006.

Received 30 March, 2005;

accepted 03 February, 2006.

Communicated by:J. Sándor

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Journal of Inequalities in Pure and Applied Mathematics

A NOTE ON MULTIPLICATIVELYe-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

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2000Victoria University ISSN (electronic): 1443-5756 313-05

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A Note on Multiplicatively e-perfect Numbers

Le Anh Vinh and Dang Phuong Dung

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Abstract

LetT_e(n)denote the product of all exponential divisors ofn. An integernis called multiplicativelye-perfect ifTe(n) =n²and multiplicativelye-superperfect ifTe(Te(n)) =n². In this note, we give an alternative proof for characterization of multiplicativelye-perfect and multiplicativelye-superperfect numbers.

2000 Mathematics Subject Classification:11A25, 11A99.

Key words: Perfect number, Exponential divisor, Multiplicatively perfect, Sum of divi- sors, Number of divisors.

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 ifσ(σ(n)) = 2n. Ifn = p^α₁¹· · ·p^α_k^k is the prime factorization ofn >1, a divisord | n, called an exponential divisor (e-divisor) ofnisd =p^β₁¹· · ·p^β_k^k withβ_i |α_ifor all1≤i≤k. LetT_e(n)denote the product of all exponential divisors ofn. The concepts of multiplicatively e-perfect and multiplicativelye-superperfect numbers were first introduced by Sándor in [1].

Definition 1.1. An integer n is called multiplicatively e-perfect if Te(n) = n² and multiplicativelye-superperfect ifT_e(T_e(n)) =n².

In [1], Sándor completely characterizes multiplicativelye-perfect and multiplicativelye-superperfect numbers.

Theorem 1.1 ([1]).

a) An integern is multiplicativelye-perfect if and only ifn = p^α, wherepis prime andαis a perfect number.

b) An interger n is multiplicatively e-superperfect if and only if n = p^α, wherepis a prime, andαis a superperfect number.

Sándor’s proof is based on an explicit expression ofT_e(n). In this note, we offer an alternative proof of Theorem1.1.

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2. Proof of Theorem 1.1

a) Suppose that n is multiplicatively e-perfect; that is T_e(n) = n². If n has more than one prime factor then n = p^α₁¹· · ·p^α_k^k for somek ≥ 2, αi ≥ 1 and p₁, . . . , p_k arekdistinct primes. We have three separate cases.

1. Suppose thatα₁ =· · ·=α_k = 1. Thendis an exponential divisor ofnif and only ifd=p1· · ·pk =n. HenceTe(n) =n,which is a contradiction.

2. Suppose that two of α₁, . . . , α_k are greater 1. Without loss of generality, we may assume that α₁, α₂ > 1. Then d₁ = p₁p^α₂²· · ·p^α_k^k, d₂ = p^α₁¹p₂p^α₃³· · ·p^α_k^k,d₃ =nare three distinct exponential divisors ofn. Hence d₁d₂d₃ |T_e(n). However,p^2α₁ ¹⁺¹ |d₁d₂d₃ soT_e(n)6=n²,which is a contradiction.

3. Suppose that there is exactly one of α₁, . . . , α_k which is greater than 1.

Without loss of generality, we may assume thatα₁ > 1andα₂ = · · · = αk = 1. We have that if d is an exponential divisor of n then d = p^β₁¹p₂· · ·p_k for some β₁ | α₁. Hence if n has more than two distinct exponential divisors then p³₂ | T_e(n) = p^2α₁ ¹p²₂· · ·p²_k, which is a contradiction. However, d1 = p1p2· · ·pk, d2 = p^α₁¹p2p3· · ·pk are two distinct exponential divisors ofnsod₁, d₂are all exponential divisors ofn. Hence T_e(n) = p^α₁¹⁺¹p²₂· · ·p²_k = p^2α₁ ¹p²₂· · ·p²_k. This implies thatα₁ = 1, which is a contradiction.

Thusnhas only one prime factor; that is,n =p^α for some primep. In this

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b) Suppose that nis multiplicativelye-superperfect; that isT_e(T_e(n)) =n². If n has more than one prime factor thenn =p^α₁¹· · ·p^α_k^k for somek ≥2,αi ≥ 1 andp₁, . . . , p_karek distinct primes. We have two separate cases.

1. Suppose thatα₁ =· · ·=α_k = 1. Thendis an exponential divisor ofnif and only ifd=p₁· · ·p_k =n. HenceT_e(n) =nandT_e(T_e(n)) = T_e(n) = nwhich is a contradiction.

2. Suppose that there is at least one ofα₁, . . . , α_kwhich is greater1. Without loss of generality, we may assume thatα₁ > 1. Thend₁ = p₁p^α₂²· · ·p^α_k^k, d2 = n = p^α₁¹p^α₂²p^α₃³· · ·p^α_k^k, are two distinct exponential divisors of n.

Hence d₁d₂ | T_e(n). However, d₁d₂ = p^α₁¹⁺¹p^2α₂ ²· · ·p^2α_k ^k so T_e(n) = p^γ₁¹· · ·p^γ_k^k where γ₁ ≥ α₁ + 1, γ_i ≥ 2α_i ≥ 2 for i = 2, . . . , k. Thus, t1 = p^γ₁¹p2p^γ₃³· · ·p^γ_k^k andt2 = Te(n) = p^γ₁¹p^γ₂²p^γ₃³· · ·p^γ_k^k are two distinct exponential divisors ofT_e(n). Hence t₁t₂ | T_e(T_e(n)). However, p^2γ₁ ¹ | t₁t₂andγ₁ > α₁,which is a contradiction.

Thusnhas only one prime factor; that is n = p^α for some prime p. In this case then T_e(n) = p^σ(α) andT_e(T_e(n)) = p^σ(σ(n)). HenceT_e(T_e(n)) = n² = p^2αif and only ifσ(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 T_e(n), as well as connected notions one-perfect numbers and generalizations. These are [2], [3], and [4].

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References

[1] J. SÁNDOR, On multiplicativelye-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 func- tions, 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]