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On Abundancy Index of Some Special Class of Numbers
Sohag J. Math. 4, No. 2, 37-40 (2017) 37 Sohag Journal of Mathematics An International Journal http://dx.doi.org/10.18576/sjm/040202 On Abundancy Index of Some Special Class of Numbers Bhabesh Das1,∗ and Helen K. Saikia2 1 Department of Mathematics, B.P.C.College, Assam, 781127, India 2 Department of Mathematics, Gauhati University, Assam, 781014, India Received: 10 Dec. 2015, Revised: 12 Sep. 2016, Accepted: 23 Sep. 2016 Published online: 1 May 2017 σ(n) σ Abstract: For any positive integer n, abundancy index I(n) is defined as I(n) = n , where (n) is the sum of all positive divisors of n. In this paper, we have discussed non trivial lower and upper bounds of I(n) for some special class of numbers like Quasi perfect numbers, Super hyperperfect numbers, Near perfect numbers and Hyperperfect numbers. Keywords: Quasi perfect number, Super hyperperfect number, Near perfect number, Hyperperfect number p−1 1 Introduction of n. If m = 2 Mp is an even perfect number, where p p both p and Mp = 2 − 1 are primes, then n = 2m, n = 2 m We known divisor function σ(n) is the sum of all positive and n = (2p − 1)m are near perfect numbers. There are divisors of n , including 1 and n itself. The abundancy finitely many near perfect numbers other than these three p index I(n) for any positive integer n is associated with the shapes. The primes of the form Mp = 2 − 1 are called divisor function σ(n) and is defined as Mersenne primes. -
Generalized Perfect Numbers
Acta Univ. Sapientiae, Mathematica, 1, 1 (2009) 73–82 Generalized perfect numbers Antal Bege Kinga Fogarasi Sapientia–Hungarian University of Sapientia–Hungarian University of Transilvania Transilvania Department of Mathematics and Department of Mathematics and Informatics, Informatics, Tˆargu Mure¸s, Romania Tˆargu Mure¸s, Romania email: [email protected] email: [email protected] Abstract. Let σ(n) denote the sum of positive divisors of the natural number n. A natural number is perfect if σ(n) = 2n. This concept was already generalized in form of superperfect numbers σ2(n)= σ(σ(n)) = k+1 k−1 2n and hyperperfect numbers σ(n)= k n + k . In this paper some new ways of generalizing perfect numbers are inves- tigated, numerical results are presented and some conjectures are estab- lished. 1 Introduction For the natural number n we denote the sum of positive divisors by arXiv:1008.0155v1 [math.NT] 1 Aug 2010 σ(n)= X d. d|n Definition 1 A positive integer n is called perfect number if it is equal to the sum of its proper divisors. Equivalently: σ(n)= 2n, where AMS 2000 subject classifications: 11A25, 11Y70 Key words and phrases: perfect number, superperfect number, k-hyperperfect number 73 74 A. Bege, K. Fogarasi Example 1 The first few perfect numbers are: 6, 28, 496, 8128, . (Sloane’s A000396 [15]), since 6 = 1 + 2 + 3 28 = 1 + 2 + 4 + 7 + 14 496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 Euclid discovered that the first four perfect numbers are generated by the for- mula 2n−1(2n − 1). -
Mathematical Gems
AMS / MAA DOLCIANI MATHEMATICAL EXPOSITIONS VOL 1 Mathematical Gems ROSS HONSBERGER MATHEMATICAL GEMS FROM ELEMENTARY COMBINATORICS, NUMBER THEORY, AND GEOMETRY By ROSS HONSBERGER THE DOLCIANI MATHEMATICAL EXPOSITIONS Published by THE MArrHEMATICAL ASSOCIATION OF AMERICA Committee on Publications EDWIN F. BECKENBACH, Chairman 10.1090/dol/001 The Dolciani Mathematical Expositions NUMBER ONE MATHEMATICAL GEMS FROM ELEMENTARY COMBINATORICS, NUMBER THEORY, AND GEOMETRY By ROSS HONSBERGER University of Waterloo Published and Distributed by THE MATHEMATICAL ASSOCIATION OF AMERICA © 1978 by The Mathematical Association of America (Incorporated) Library of Congress Catalog Card Number 73-89661 Complete Set ISBN 0-88385-300-0 Vol. 1 ISBN 0-88385-301-9 Printed in the United States of Arnerica Current printing (last digit): 10 9 8 7 6 5 4 3 2 1 FOREWORD The DOLCIANI MATHEMATICAL EXPOSITIONS serIes of the Mathematical Association of America came into being through a fortuitous conjunction of circumstances. Professor-Mary P. Dolciani, of Hunter College of the City Uni versity of New York, herself an exceptionally talented and en thusiastic teacher and writer, had been contemplating ways of furthering the ideal of excellence in mathematical exposition. At the same time, the Association had come into possession of the manuscript for the present volume, a collection of essays which seemed not to fit precisely into any of the existing Associa tion series, and yet which obviously merited publication because of its interesting content and lucid expository style. It was only natural, then, that Professor Dolciani should elect to implement her objective by establishing a revolving fund to initiate this series of MATHEMATICAL EXPOSITIONS. -
On Perfect Numbers and Their Relations 1 Introduction
Int. J. Contemp. Math. Sciences, Vol. 5, 2010, no. 27, 1337 - 1346 On Perfect Numbers and their Relations Recep G¨ur 2000 Evler Mahallesi Boykent Sitesi E Blok 26/7 Nevsehir, Turkey [email protected] Nihal Bircan Technische Universitaet Berlin Fakultaet II Institut f¨ur Mathematik MA 8 − 1 Strasse des 17. Juni 136 D-10623 Berlin, Germany [email protected] Abstract In this paper, we get some new formulas for generalized perfect numbers and their relationship between arithmetical functions φ, σ concerning Ore’s harmonic numbers and by using these formulas we present some examples. Mathematics Subject Classification: 11A25, 11A41, 11Y70 Keywords: perfect number, 2-hyperperfect number, Euler’s totient function, Ore harmonic number 1 Introduction In this section, we aimed to provide general information on perfect numbers shortly. Here and throughout the paper we assume m, n, k, d, b are positive integers and p is prime. N is called perfect number, if it is equal to the sum of its proper divisors. The first few perfect numbers are 6, 28, 496, 8128,... since, 6 = 1+2+3 28 = 1+2+4+7+14 496 = 1+2+4+8+16+31+62+124+248 Euclid discovered the first four perfect numbers which are generated by the formula 2n−1(2n−1), called Euclid number. In his book ’Elements’ he presented the proof of the formula which gives an even perfect number whenever 2n −1is 1338 Recep G¨ur and Nihal Bircan prime. In order for 2n −1 to be a prime n must itself be a prime. -
Identities for Near and Deficient Hyperperfect Numbers
Received: 03.01.2016 Year: 2017, Pages: 01-12 Published: 13.04.2017 Original Article** Identities for Near and Deficient Hyperperfect Numbers Bhabesh Das1;* <[email protected]> Helen K.Saikia2 <[email protected]> 1Department of Mathematics,B.P.C.College,781127,Assam,India 2Department of Mathematics,Gauhati University,781014,Assam,India Abstaract − It is well-known that a positive integer n is said to be k−hyperperfect number if k + 1 k − 1 σ(n) = n + ; k k where k is a positive integer and σ(n) is the sum of all positive divisors of n. We call a number n is near k−hyperperfect number if k + 1 k − 1 σ(n) = n + + d k k and decient k−hyperperfect number if k + 1 k − 1 σ(n) = n + − d; k k where d is a proper divisor of n. In this paper, for any prime number q, we present two classes of near (q − 1)−hyperperfect number and one class of decient (q − 1)−hyperperfect number with two distinct prime factors and also present some numerical results. Keywords − Perfect number, Hyperperfect number, Near perfect number, Decient perfect number. **Edited by A. D. Godase (Editor-in-Chief). *Corresponding Author. Indian Journal in Number Theory, 2017, 01-12 2 1 Introduction For any positive integer n, divisor function σ(n) denote the sum of all positive divisors of n, X σ(n) = d: djn Denition 1.1. A positive integer n is called perfect number, if σ(n) = 2n. All known perfect numbers are even. The Euclid-Euler theorem gives the general form of even perfect numbers. -
POWERFUL AMICABLE NUMBERS 1. Introduction Let S(N) := ∑ D Be the Sum of the Proper Divisors of the Natural Number N. Two Disti
POWERFUL AMICABLE NUMBERS PAUL POLLACK P Abstract. Let s(n) := djn; d<n d denote the sum of the proper di- visors of the natural number n. Two distinct positive integers n and m are said to form an amicable pair if s(n) = m and s(m) = n; in this case, both n and m are called amicable numbers. The first example of an amicable pair, known already to the ancients, is f220; 284g. We do not know if there are infinitely many amicable pairs. In the opposite direction, Erd}osshowed in 1955 that the set of amicable numbers has asymptotic density zero. Let ` ≥ 1. A natural number n is said to be `-full (or `-powerful) if p` divides n whenever the prime p divides n. As shown by Erd}osand 1=` Szekeres in 1935, the number of `-full n ≤ x is asymptotically c`x , as x ! 1. Here c` is a positive constant depending on `. We show that for each fixed `, the set of amicable `-full numbers has relative density zero within the set of `-full numbers. 1. Introduction P Let s(n) := djn; d<n d be the sum of the proper divisors of the natural number n. Two distinct natural numbers n and m are said to form an ami- cable pair if s(n) = m and s(m) = n; in this case, both n and m are called amicable numbers. The first amicable pair, 220 and 284, was known already to the Pythagorean school. Despite their long history, we still know very little about such pairs. -
On the Number of Distinct Exponents in the Prime Factorization of an Integer
Proc. Indian Acad. Sci. (Math. Sci.) (2020) 130:27 https://doi.org/10.1007/s12044-020-0556-y On the number of distinct exponents in the prime factorization of an integer CARLO SANNA Department of Mathematics, Università degli Studi di Genova, Genoa, Italy E-mail: [email protected] MS received 25 February 2019; revised 5 September 2019; accepted 11 November 2019 Abstract. Let f (n) be the number of distinct exponents in the prime factorization of the natural number n. We prove some results about the distribution of f (n). In particular, for any positive integer k, we obtain that #{n ≤ x : f (n) = k}∼Ak x and Bx(log log x)k #{n ≤ x : f (n) = ω(n) − k}∼ , k! log x as x →+∞,whereω(n) is the number of prime factors of n and Ak, B > 0 are some explicit constants. The latter asymptotic extends a result of Akta¸s and Ram Murty (Proc. Indian Acad. Sci. (Math. Sci.) 127(3) (2017) 423–430) about numbers having mutually distinct exponents in their prime factorization. Keywords. Prime factorization; squarefree numbers; powerful number. 2010 Mathematics Subject Classification. Primary: 11N25; Secondary: 11N37, 11N64. 1. Introduction = a1 ··· as > < Let n p1 ps be the factorization of the natural number n 1, where p1 ··· < ps are prime numbers and a1,...,as are positive integers. Several functions of the exponents a1,...,as have been studied, including their product [17], their arithmetic mean [2,4,5,7], and their maximum and minimum [11,13,15,18]. See also [3,8]fora more general function. -
Some Problems in Partitio Numerorum
J. Austral. Math. Soc. (Series A) 27 (1979), 319-331 SOME PROBLEMS IN PARTITIO NUMERORUM P. ERDOS and J. H. LOXTON (Received 24 April 1977) Communicated by J. Pitman Abstract We consider some unconventional partition problems in which the parts of the partition are restricted by divisibility conditions, for example, partitions n = ax +... + a* into positive integers «!, ..., ak such that ax | a2 I ••• I ak. Some rather weak estimates for the various partition functions are obtained. Subject classification (Amer. Math. Soc. (MOS) 1970): 10 A 45, 10 J 20. 1. Introduction In this paper, we shall consider various partition problems in which the parts of the partitions are restricted by divisibility conditions. Most of our remarks concern the following two situations: (i) 'Chain partitions', that is partitions n = a1 + ... +ak into positive integers a1,...,ak such that a^a^ ...\ak. (ii) 'Umbrella partitions', that is partitions into positive integers such that every part divides the largest one. Our aim is to estimate the partition functions which arise in each case for partitions with distinct parts and for partitions in which repetitions are allowed. This work arose from a question of R. W. Robinson about chain partitions with repetitions which, in turn, came from attempts to count a certain kind of tree. This particular partition problem is closely connected with wj-ary partitions, that is partitions as sums of powers of a fixed integer m, which are obvious instances of the types of partitions described above. In another direction, the problem of representing numbers by umbrella partitions has some connections with the 'practical numbers' of Srinivasan. -
Integer Sequences
UHX6PF65ITVK Book > Integer sequences Integer sequences Filesize: 5.04 MB Reviews A very wonderful book with lucid and perfect answers. It is probably the most incredible book i have study. Its been designed in an exceptionally simple way and is particularly just after i finished reading through this publication by which in fact transformed me, alter the way in my opinion. (Macey Schneider) DISCLAIMER | DMCA 4VUBA9SJ1UP6 PDF > Integer sequences INTEGER SEQUENCES Reference Series Books LLC Dez 2011, 2011. Taschenbuch. Book Condition: Neu. 247x192x7 mm. This item is printed on demand - Print on Demand Neuware - Source: Wikipedia. Pages: 141. Chapters: Prime number, Factorial, Binomial coeicient, Perfect number, Carmichael number, Integer sequence, Mersenne prime, Bernoulli number, Euler numbers, Fermat number, Square-free integer, Amicable number, Stirling number, Partition, Lah number, Super-Poulet number, Arithmetic progression, Derangement, Composite number, On-Line Encyclopedia of Integer Sequences, Catalan number, Pell number, Power of two, Sylvester's sequence, Regular number, Polite number, Ménage problem, Greedy algorithm for Egyptian fractions, Practical number, Bell number, Dedekind number, Hofstadter sequence, Beatty sequence, Hyperperfect number, Elliptic divisibility sequence, Powerful number, Znám's problem, Eulerian number, Singly and doubly even, Highly composite number, Strict weak ordering, Calkin Wilf tree, Lucas sequence, Padovan sequence, Triangular number, Squared triangular number, Figurate number, Cube, Square triangular -
On Zumkeller Numbers
On Zumkeller Numbers K.P.S. Bhaskara Rao Yuejian Peng Department of Mathematics and Computer Science Indiana State University Terre Haute, IN, 47809, USA Email: [email protected]; [email protected] Abstract Generalizing the concept of a perfect number, Sloane’s sequences of integers A083207 lists the sequence of integers n with the property: the positive factors of n can be partitioned into two disjoint parts so that the sums of the two parts are equal. Following [4] Clark et al., we shall call such integers, Zumkeller numbers. Generalizing this, in [4] a number n is called a half-Zumkeller number if the positive proper factors of n can be partitioned into two disjoint parts so that the sums of the two parts are equal. An extensive study of properties of Zumkeller numbers, half-Zumkeller numbers and their relation to practical numbers is undertaken in this paper. In [4] Clark et al., announced results about Zumkellers numbers and half- Zumkeller numbers and suggested two conjectures. In the present paper we shall settle one of the conjectures, prove the second conjecture in some special cases and arXiv:0912.0052v1 [math.NT] 1 Dec 2009 prove several results related to the second conjecture. We shall also show that if there is an even Zumkeller number that is not half-Zumkeller it should be bigger than 7.2334989 × 109. 1 Introduction A positive integer n is called a perfect number if n equals the sum of its proper positive factors. Generalizing this concept in 2003, Zumkeller published in Sloane’s sequences of inte- gers A083207 a sequence of integers n with the property that the positive factors of n 1 can be partitioned into two disjoint parts so that the sums of the two parts are equal. -
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 ............................... -
On Near 3−Perfect Numbers -.:: Natural Sciences Publishing
Sohag J. Math. 4, No. 1, 1-5 (2017) 1 Sohag Journal of Mathematics An International Journal http://dx.doi.org/10.18576/sjm/040101 On Near 3−Perfect Numbers Bhabesh Das1,∗ and Helen K. Saikia2 1 Department of Mathematics, B.P.C.College, Assam, 781127, India 2 Department of Mathematics, Gauhati University, Assam, 781014, India Received: 10 Jan. 2016, Revised: 18 Sep. 2016, Accepted: 20 Sep. 2016 Published online: 1 Jan. 2017 Abstract: We call a positive integer n be a near 3−perfect number if σ(n) = 3n + d, where σ(n) is the divisor function and d is a α t proper divisor of n. In this paper, we have derived all near 3−perfects of the form 2 p1 p2, where p1 and p2 are distinct odd primes with p1 < p2 and α ≥ 1, 1 ≤ t ≤ 2. There are only ten such numbers. Moreover, we have also obtained some examples of even near 3−perfect numbers with four distinct prime factors. Keywords: Multi perfect number, Hyperperfect number, Near perfect number 1 Introduction The abudancy index I(n) for any positive integers n is associated with the divisor function σ(n) and is defined as σ(n) I(n)= n . Since k−perfect numbers are solutions of the Well known divisor function σ(n) is the sum of all positive equation σ(n)= kn, so I(n)= k. For abundant numbers, divisors of n including 1 and n itself. For any integer k > 1, I(n) > 2 and for deficient numbers, I(n) < 2. a positive integer n is called a k−perfect or multi perfect For any positive integer k > 1, we call a number n be number [3,7], if σ(n)= kn.