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Some Results on Generalized Multiplicative Perfect Numbers Alexandre Laugier Lyc´eeprofessionnel Tristan Corbi`ere 16 rue de Kerveguen - BP 17149 29671 Morlaix cedex France [email protected] Manjil P. Saikia1 Fakult¨atf¨urMathematik Universit¨atWien Oskar-Morgenstern-Platz 1 1090 Wien Austria [email protected] Abstract In this article, we define k-multiplicatively e-perfect numbers and k-multiplicatively e-superperfect numbers and prove some results on them. We also characterize the k- ∗ T0T -perfect numbers defined by Das and Saikia (2013). 1 Introduction A natural number n is said to be perfect (A000396) if the sum of all aliqout divisors of n is equal to n. Or equivalently, σ(n) = 2n, where σ(k) is the sum of the divisors of k. It is a well known result of Euler-Euclid that the form of even perfect numbers is n = 2kp, where p = 2k+1 − 1 is a Mersenne prime and k ≥ 1. Till date, no odd perfect number is known, and it is believed that none exists. Moreover n is said to be super-perfect if σ(σ(n)) = 2n. It was proved by Suryanarayana-Kanold [3,7] that the general form of such super-perfect numbers are n = 2k, where 2k+1 − 1 is a prime and k ≥ 1. No odd super-perfect numbers are known till date. Unless, otherwise mentioned all n considered in this paper will be a natural number. We also denote by 1; n , the set f1; 2; : : : ; ng. J K 1Corresponding author 1 Let T (n) denote the product of all the divisors of n.A multiplicatively perfect (A007422) number is a number n such that T (n) = n2 and n is called a multiplicatively super-perfect if T (T (n)) = n2. S´andor [4] characterized such numbers and also numbers called k-multiplicatively perfect numbers, which are numbers n, such that T (n) = nk for k ≥ 2. In this article we shall give some results on other classes of perfect numbers defined by various authors. For a general introduction to such numbers, we refer the readers to the article [2], which contains various references to the existing literature. 2 k-multiplicatively e-perfect numbers S´andor[5] studied the multiplicatively e-perfect numbers defined in the previous section. If a1 ar n = p1 ··· pr is the prime factorization of n > 1, a divisor d of n is called an exponential b1 br divisor (or, e-divisor for short) if d = p1 ··· pr with bi j ai for i = 1; : : : ; r. This notion is due to Straus and Subbarao [6]. Let σe(n) denote the sum of e-divisors of n, then Straus and Subbarao define n as exponentially perfect (or, e-perfect for short) (A054979) if σe(n) = 2n. They proved that there are no odd e-perfect numbers, and for each r, such numbers with r prime factors are finite. We refer the reader to the article [5] for some historical comments and results related to such e-perfect numbers. Let Te(n) denote the product of all the e-divisors of n. Then n is called multiplicatively 2 2 e-perfect if Te(n) = n and multiplicatively e-superperfect if Te(Te(n)) = n . The main result of S´andor[5] is the following. Theorem 1 (S´andor,[5] Theorem 2.1). n is multiplicatively e-perfect if and only if n = pa, where p is a prime and a is an ordinary perfect number. n is multiplicatively e-superperfect if and only if n = pa, where p is a prime and a is an ordinary superperfect number, that is σ(σ(a)) = 2a. Inspired by the work of others in introducing generalized multiplicative perfect numbers, we now introduce the following two classes of numbers. k Definition 2. A natural number n is called k-multiplicatively e-perfect if Te(n) = n , where k ≥ 2. Definition 3. A natural number n is called a k-multiplicatively e-superperfect number k if Te(Te(n)) = n , where k ≥ 2. Example 4. There exist 6k-multiplicatively e-perfect numbers with k 2 N?, which have the α form (p1 · p2 · p3) provided σ(α)d(α)2 = 6kα Indeed, according to Theorem 2.1 of [5], we have α σ(α)d(α)2 Te((p1 · p2 · p3) ) = (p1 · p2 · p3) Particularly, if d(α) = 6, then αk = 6 · σ(α). Some cases are listed below 2 • α = 2, k = 1 where d(α) = 2 and σ(α) = 3; • α = 8, k = 5 where d(α) = 4 and σ(α) = 15; • α = 28, k = 12 where d(α) = 6 and σ(α) = 56; • α = 18, k = 13 where d(α) = 6 and σ(α) = 39; • α = 12, k = 14 where d(α) = 6 and σ(α) = 28. Afterwards, we characterize some of these class of numbers in the following theorems. Theorem 5. If n = pa, where p is a prime and a is a k-perfect number, then n is k- multiplicatively e-perfect. If n = pa, where p is a prime and a is a k-superperfect number, then n is k-multiplicatively e-superperfect. Proof. We know from S´andor[5], that if n = pa where p is a prime and a is a non-zero positive integer, then we have σ(a) Te(n) = p : So, if n = pa where p is a prime and a is a k-perfect number, then σ(a) = ka; and so ka k Te(n) = p = n : Moreover, if n = pa where p is a prime and a is a non-zero positive integer, then we have σ(a) σ(σ(a)) Te(Te(n)) = Te(p ) = p : So, if n = pa where p is a prime and a is a k-perfect number, then σ(σ(a)) = ka; and so ka k Te(Te(n)) = p = n : In the following, we denote by d(n), the number of positive divisors of n. α1 αr ∗ Theorem 6. Let p be a prime number and let n = p1 ··· pr , with r 2 N be the prime ∗ factorisation of an integer n > 1 where pi with i 2 1; r are prime numbers and αi 2 N for all i 2 1; r . n is p-multiplicatively e-perfect if andJ onlyK if for each i 2 1; r , we have J K J K σ(αi) = gcd(αi; σ(αi))p 3 and Y αi = gcd(αi; σ(αi)) d(αj); j2 1;r nfig J K with r−1 Y 2 ≤ d(αj) < p; j2 1;r nfig J K where we set Y d(αj) = 1: j2; In particular, if r = 1, then α1jσ(α1) and we have σ(α1) = α1p: Proof. Before proceeding to the proof, we know from S´andor[5] that σ(α1)d(α2)···d(αr) σ(αr)d(α1)···d(αr−1) Te(n) = p1 ··· pr : (2.1) α1 If r = 1, then using Theorem5, n = p1 is p-multiplivatively e-perfect if and only if α1 is a p-perfect number. We now assume that r ≥ 2. We have 8 > σ(α1)d(α2) ··· d(αr) = pα1; > p < Te(n) = n , ······ (2.2) > > : σ(αr)d(α1) ··· d(αr−1) = pαr: Clearly, if for each i 2 1; r , we have J K σ(αi) = gcd(αi; σ(αi))p and Y αi = gcd(αi; σ(αi)) d(αj) j2 1;r nfig J K then it can be easily verified that (2.2) is satisfied. Now we notice that if all αi (i 2 1; r ) are equal to 1, then (2.2) is consistent only if p = 1. It is impossible since p is a primeJ number.K If at least an αi is equal to 1, say α1 = 1, then from (2.2), we have d(α2) ··· d(αr) = p: Then one of d(α2); : : : ; d(αr) is equal to the prime p and the others are equal to 1. Say d(α2) = p; 4 and d(α3) = ··· = d(αr) = 1: It implies that α1 = α3 = ··· = αr = 1: Then the equation σ(α2)d(α1)d(α3) ··· d(αr) = pα2 of (2.2) gives σ(α2) = pα2. But, for all i 2 1; r n f1; 2g, the equation σ(αi)d(α1)d(α2) ··· d(αr) = pαi of (2.2) gives σ(αi) = αi whichJ isK not possible since we know that σ(αi) ≥ αi + 1. So, we must have αi ≥ 2 for all i 2 1; r . Thus, J K Y r−1 d(αj) ≥ 2 : j2 1;r nfig J K We now prove that if (2.2) is true, then for each i 2 1; r , we have J K σ(αi) = gcd(αi; σ(αi))p and Y αi = gcd(αi; σ(αi)) d(αj): j2 1;r nfig J K Let gi = gcd(αi; σ(αi)) for all i 2 1; r . So, for each i 2 1; r , there exists two non-zero positive integer ai; si such that αi =J giaKi and σ(αi) = gisi withJ K gcd(ai; si) = 1. Notice that si > 1. Otherwise, we would have σ(αi)jαi, a contradiction. For each i 2 1; r , the equation J K Y σ(αi) d(αj) = pαi j2 1;r nfig J K of (2.2) now gives Y si d(αj) = pai: j2 1;r nfig J K Since gcd(ai; si) = 1, then from Euclid's lemma we get Y aij d(αj); j2 1;r nfig J K and sijp: So, there exists an integer k such that Y d(αj) = kai; j2 1;r nfig J K and p = ksi: 5 Using the fact that p is a prime and si > 1, it results that k = 1 and we get Y d(αj) = ai; j2 1;r nfig J K and p = si: Therefore σ(αi) = gip and gcd(p; ai) = 1.
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  • Handbook of Number Theory Ii

    Handbook of Number Theory Ii

    HANDBOOK OF NUMBER THEORY II by J. Sandor´ Babes¸-Bolyai University of Cluj Department of Mathematics and Computer Science Cluj-Napoca, Romania and B. Crstici formerly the Technical University of Timis¸oara Timis¸oara Romania KLUWER ACADEMIC PUBLISHERS DORDRECHT / BOSTON / LONDON A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN 1-4020-2546-7 (HB) ISBN 1-4020-2547-5 (e-book) Published by Kluwer Academic Publishers, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. Sold and distributed in North, Central and South America by Kluwer Academic Publishers, 101 Philip Drive, Norwell, MA 02061, U.S.A. In all other countries, sold and distributed by Kluwer Academic Publishers, P.O. Box 322, 3300 AH Dordrecht, The Netherlands. Printed on acid-free paper All Rights Reserved C 2004 Kluwer Academic Publishers No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed in the Netherlands. Contents PREFACE 7 BASIC SYMBOLS 9 BASIC NOTATIONS 10 1 PERFECT NUMBERS: OLD AND NEW ISSUES; PERSPECTIVES 15 1.1 Introduction .............................. 15 1.2 Some historical facts ......................... 16 1.3 Even perfect numbers ......................... 20 1.4 Odd perfect numbers ......................... 23 1.5 Perfect, multiperfect and multiply perfect numbers ......... 32 1.6 Quasiperfect, almost perfect, and pseudoperfect numbers ...............................