On Modified Hyperperfect Numbers
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Bi-Unitary Multiperfect Numbers, I
Notes on Number Theory and Discrete Mathematics Print ISSN 1310–5132, Online ISSN 2367–8275 Vol. 26, 2020, No. 1, 93–171 DOI: 10.7546/nntdm.2020.26.1.93-171 Bi-unitary multiperfect numbers, I Pentti Haukkanen1 and Varanasi Sitaramaiah2 1 Faculty of Information Technology and Communication Sciences, FI-33014 Tampere University, Finland e-mail: [email protected] 2 1/194e, Poola Subbaiah Street, Taluk Office Road, Markapur Prakasam District, Andhra Pradesh, 523316 India e-mail: [email protected] Dedicated to the memory of Prof. D. Suryanarayana Received: 19 December 2019 Revised: 21 January 2020 Accepted: 5 March 2020 Abstract: A divisor d of a positive integer n is called a unitary divisor if gcd(d; n=d) = 1; and d is called a bi-unitary divisor of n if the greatest common unitary divisor of d and n=d is unity. The concept of a bi-unitary divisor is due to D. Surynarayana [12]. Let σ∗∗(n) denote the sum of the bi-unitary divisors of n: A positive integer n is called a bi-unitary perfect number if σ∗∗(n) = 2n. This concept was introduced by C. R. Wall in 1972 [15], and he proved that there are only three bi-unitary perfect numbers, namely 6, 60 and 90. In 1987, Peter Hagis [6] introduced the notion of bi-unitary multi k-perfect numbers as solu- tions n of the equation σ∗∗(n) = kn. A bi-unitary multi 3-perfect number is called a bi-unitary triperfect number. A bi-unitary multiperfect number means a bi-unitary multi k-perfect number with k ≥ 3: Hagis [6] proved that there are no odd bi-unitary multiperfect numbers. -
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). -
ON PERFECT and NEAR-PERFECT NUMBERS 1. Introduction a Perfect
ON PERFECT AND NEAR-PERFECT NUMBERS PAUL POLLACK AND VLADIMIR SHEVELEV Abstract. We call n a near-perfect number if n is the sum of all of its proper divisors, except for one of them, which we term the redundant divisor. For example, the representation 12 = 1 + 2 + 3 + 6 shows that 12 is near-perfect with redundant divisor 4. Near-perfect numbers are thus a very special class of pseudoperfect numbers, as defined by Sierpi´nski. We discuss some rules for generating near-perfect numbers similar to Euclid's rule for constructing even perfect numbers, and we obtain an upper bound of x5=6+o(1) for the number of near-perfect numbers in [1; x], as x ! 1. 1. Introduction A perfect number is a positive integer equal to the sum of its proper positive divisors. Let σ(n) denote the sum of all of the positive divisors of n. Then n is a perfect number if and only if σ(n) − n = n, that is, σ(n) = 2n. The first four perfect numbers { 6, 28, 496, and 8128 { were known to Euclid, who also succeeded in establishing the following general rule: Theorem A (Euclid). If p is a prime number for which 2p − 1 is also prime, then n = 2p−1(2p − 1) is a perfect number. It is interesting that 2000 years passed before the next important result in the theory of perfect numbers. In 1747, Euler showed that every even perfect number arises from an application of Euclid's rule: Theorem B (Euler). All even perfect numbers have the form 2p−1(2p − 1), where p and 2p − 1 are primes. -
On the Largest Odd Component of a Unitary Perfect Number*
ON THE LARGEST ODD COMPONENT OF A UNITARY PERFECT NUMBER* CHARLES R. WALL Trident Technical College, Charleston, SC 28411 (Submitted September 1985) 1. INTRODUCTION A divisor d of an integer n is a unitary divisor if gcd (d9 n/d) = 1. If d is a unitary divisor of n we write d\\n9 a natural extension of the customary notation for the case in which d is a prime power. Let o * (n) denote the sum of the unitary divisors of n: o*(n) = £ d. d\\n Then o* is a multiplicative function and G*(pe)= 1 + p e for p prime and e > 0. We say that an integer N is unitary perfect if o* (N) = 2#. In 1966, Sub- baro and Warren [2] found the first four unitary perfect numbers: 6 = 2 * 3 ; 60 = 223 - 5 ; 90 = 2 * 325; 87,360 = 263 • 5 • 7 • 13. In 1969s I announced [3] the discovery of another such number, 146,361,936,186,458,562,560,000 = 2183 • 5^7 • 11 • 13 • 19 • 37 • 79 • 109 * 157 • 313, which I later proved [4] to be the fifth unitary perfect number. No other uni- tary perfect numbers are known. Throughout what follows, let N = 2am (with m odd) be unitary perfect and suppose that K is the largest odd component (i.e., prime power unitary divisor) of N. In this paper we outline a proof that, except for the five known unitary perfect numbers, K > 2 2. TECHNIQUES In light of the fact that 0*(pe) = 1 + pe for p prime, the problem of find- ing a unitary perfect number is equivalent to that of expressing 2 as a product of fractions, with each numerator being 1 more than its denominator, and with the denominators being powers of distinct primes. -
PRIME-PERFECT NUMBERS Paul Pollack [email protected] Carl
PRIME-PERFECT NUMBERS Paul Pollack Department of Mathematics, University of Illinois, Urbana, Illinois 61801, USA [email protected] Carl Pomerance Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755, USA [email protected] In memory of John Lewis Selfridge Abstract We discuss a relative of the perfect numbers for which it is possible to prove that there are infinitely many examples. Call a natural number n prime-perfect if n and σ(n) share the same set of distinct prime divisors. For example, all even perfect numbers are prime-perfect. We show that the count Nσ(x) of prime-perfect numbers in [1; x] satisfies estimates of the form c= log log log x 1 +o(1) exp((log x) ) ≤ Nσ(x) ≤ x 3 ; as x ! 1. We also discuss the analogous problem for the Euler function. Letting N'(x) denote the number of n ≤ x for which n and '(n) share the same set of prime factors, we show that as x ! 1, x1=2 x7=20 ≤ N (x) ≤ ; where L(x) = xlog log log x= log log x: ' L(x)1=4+o(1) We conclude by discussing some related problems posed by Harborth and Cohen. 1. Introduction P Let σ(n) := djn d be the sum of the proper divisors of n. A natural number n is called perfect if σ(n) = 2n and, more generally, multiply perfect if n j σ(n). The study of such numbers has an ancient pedigree (surveyed, e.g., in [5, Chapter 1] and [28, Chapter 1]), but many of the most interesting problems remain unsolved. -
Types of Integer Harmonic Numbers (Ii)
Bulletin of the Transilvania University of Bra¸sov • Vol 9(58), No. 1 - 2016 Series III: Mathematics, Informatics, Physics, 67-82 TYPES OF INTEGER HARMONIC NUMBERS (II) Adelina MANEA1 and Nicu¸sorMINCULETE2 Abstract In the first part of this paper we obtained several bi-unitary harmonic numbers which are higher than 109, using the Mersenne prime numbers. In this paper we investigate bi-unitary harmonic numbers of some particular forms: 2k · n, pqt2, p2q2t, with different primes p, q, t and a squarefree inte- ger n. 2010 Mathematics Subject Classification: 11A25. Key words: harmonic numbers, bi-unitary harmonic numbers. 1 Introduction The harmonic numbers introduced by O. Ore in [8] were named in this way by C. Pomerance in [11]. They are defined as positive integers for which the harmonic mean of their divisors is an integer. O. Ore linked the perfect numbers with the harmonic numbers, showing that every perfect number is harmonic. A list of the harmonic numbers less than 2 · 109 is given by G. L. Cohen in [1], finding a total of 130 of harmonic numbers, and G. L. Cohen and R. M. Sorli in [2] have continued to this list up to 1010. The notion of harmonic numbers is extended to unitary harmonic numbers by K. Nageswara Rao in [7] and then to bi-unitary harmonic numbers by J. S´andor in [12]. Our paper is inspired by [12], where J. S´andorpresented a table containing all the 211 bi-unitary harmonic numbers up to 109. We extend the J. S´andors's study, looking for other bi-unitary harmonic numbers, greater than 109. -
A Study of .Perfect Numbers and Unitary Perfect
CORE Metadata, citation and similar papers at core.ac.uk Provided by SHAREOK repository A STUDY OF .PERFECT NUMBERS AND UNITARY PERFECT NUMBERS By EDWARD LEE DUBOWSKY /I Bachelor of Science Northwest Missouri State College Maryville, Missouri: 1951 Master of Science Kansas State University Manhattan, Kansas 1954 Submitted to the Faculty of the Graduate College of the Oklahoma State University in partial fulfillment of .the requirements fqr the Degree of DOCTOR OF EDUCATION May, 1972 ,r . I \_J.(,e, .u,,1,; /q7Q D 0 &'ISs ~::>-~ OKLAHOMA STATE UNIVERSITY LIBRARY AUG 10 1973 A STUDY OF PERFECT NUMBERS ·AND UNITARY PERFECT NUMBERS Thesis Approved: OQ LL . ACKNOWLEDGEMENTS I wish to express my sincere gratitude to .Dr. Gerald K. Goff, .who suggested. this topic, for his guidance and invaluable assistance in the preparation of this dissertation. Special thanks.go to the members of. my advisory committee: 0 Dr. John Jewett, Dr. Craig Wood, Dr. Robert Alciatore, and Dr. Vernon Troxel. I wish to thank. Dr. Jeanne Agnew for the excellent training in number theory. that -made possible this .study. I wish tc;, thank Cynthia Wise for her excellent _job in typing this dissertation •. Finally, I wish to express gratitude to my wife, Juanita, .and my children, Sondra and David, for their encouragement and sacrifice made during this study. TABLE OF CONTENTS Chapter Page I. HISTORY AND INTRODUCTION. 1 II. EVEN PERFECT NUMBERS 4 Basic Theorems • • • • • • • • . 8 Some Congruence Relations ••• , , 12 Geometric Numbers ••.••• , , , , • , • . 16 Harmonic ,Mean of the Divisors •. ~ ••• , ••• I: 19 Other Properties •••• 21 Binary Notation. • •••• , ••• , •• , 23 III, ODD PERFECT NUMBERS . " . 27 Basic Structure • , , •• , , , . -
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. -
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 -
Some Problems of Erdős on the Sum-Of-Divisors Function
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, SERIES B Volume 3, Pages 1–26 (April 5, 2016) http://dx.doi.org/10.1090/btran/10 SOME PROBLEMS OF ERDOS˝ ON THE SUM-OF-DIVISORS FUNCTION PAUL POLLACK AND CARL POMERANCE For Richard Guy on his 99th birthday. May his sequence be unbounded. Abstract. Let σ(n) denote the sum of all of the positive divisors of n,and let s(n)=σ(n) − n denote the sum of the proper divisors of n. The functions σ(·)ands(·) were favorite subjects of investigation by the late Paul Erd˝os. Here we revisit three themes from Erd˝os’s work on these functions. First, we improve the upper and lower bounds for the counting function of numbers n with n deficient but s(n) abundant, or vice versa. Second, we describe a heuristic argument suggesting the precise asymptotic density of n not in the range of the function s(·); these are the so-called nonaliquot numbers. Finally, we prove new results on the distribution of friendly k-sets, where a friendly σ(n) k-set is a collection of k distinct integers which share the same value of n . 1. Introduction Let σ(n) be the sum of the natural number divisors of n,sothatσ(n)= d|n d. Let s(n) be the sum of only the proper divisors of n,sothats(n)=σ(n) − n. Interest in these functions dates back to the ancient Greeks, who classified numbers as deficient, perfect,orabundant according to whether s(n) <n, s(n)=n,or s(n) >n, respectively.