A Study of .Perfect Numbers and Unitary Perfect
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Special Kinds of Numbers
Chapel Hill Math Circle: Special Kinds of Numbers 11/4/2017 1 Introduction This worksheet is concerned with three classes of special numbers: the polyg- onal numbers, the Catalan numbers, and the Lucas numbers. Polygonal numbers are a nice geometric generalization of perfect square integers, Cata- lan numbers are important numbers that (one way or another) show up in a wide variety of combinatorial and non-combinatorial situations, and Lucas numbers are a slight twist on our familiar Fibonacci numbers. The purpose of this worksheet is not to give an exhaustive treatment of any of these numbers, since that is just not possible. The goal is to, instead, introduce these interesting classes of numbers and give some motivation to explore on your own. 2 Polygonal Numbers The first class of numbers we'll consider are called polygonal numbers. We'll start with some simpler cases, and then derive a general formula. Take 3 points and arrange them as an equilateral triangle with side lengths of 2. Here, I use the term \side-length" to mean \how many points are in a side." The total number of points in this triangle (3) is the second triangular number, denoted P (3; 2). To get the third triangular number P (3; 3), we take the last configuration and adjoin points so that the outermost shape is an equilateral triangle with a side-length of 3. To do this, we can add 3 points along any one of the edges. Since there are 3 points in the original 1 triangle, and we added 3 points to get the next triangle, the total number of points in this step is P (3; 3) = 3 + 3 = 6. -
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. -
Input for Carnival of Math: Number 115, October 2014
Input for Carnival of Math: Number 115, October 2014 I visited Singapore in 1996 and the people were very kind to me. So I though this might be a little payback for their kindness. Good Luck. David Brooks The “Mathematical Association of America” (http://maanumberaday.blogspot.com/2009/11/115.html ) notes that: 115 = 5 x 23. 115 = 23 x (2 + 3). 115 has a unique representation as a sum of three squares: 3 2 + 5 2 + 9 2 = 115. 115 is the smallest three-digit integer, abc , such that ( abc )/( a*b*c) is prime : 115/5 = 23. STS-115 was a space shuttle mission to the International Space Station flown by the space shuttle Atlantis on Sept. 9, 2006. The “Online Encyclopedia of Integer Sequences” (http://www.oeis.org) notes that 115 is a tridecagonal (or 13-gonal) number. Also, 115 is the number of rooted trees with 8 vertices (or nodes). If you do a search for 115 on the OEIS website you will find out that there are 7,041 integer sequences that contain the number 115. The website “Positive Integers” (http://www.positiveintegers.org/115) notes that 115 is a palindromic and repdigit number when written in base 22 (5522). The website “Number Gossip” (http://www.numbergossip.com) notes that: 115 is the smallest three-digit integer, abc, such that (abc)/(a*b*c) is prime. It also notes that 115 is a composite, deficient, lucky, odd odious and square-free number. The website “Numbers Aplenty” (http://www.numbersaplenty.com/115) notes that: It has 4 divisors, whose sum is σ = 144. -
Mathematics 2016.Pdf
UNIVERSITY INTERSCHOLASTIC LEAGUE Making a World of Diference Mathematics Invitational A • 2016 DO NOT TURN THIS PAGE UNTIL YOU ARE INSTRUCTED TO DO SO! 1. Evaluate: 1 qq 8‚ƒ 1 6 (1 ‚ 30) 1 6 (A) qq 35 (B) 11.8 (C) qqq 11.2 (D) 8 (E) 5.0333... 2. Max Spender had $40.00 to buy school supplies. He bought six notebooks at $2.75 each, three reams of paper at $1.20 each, two 4-packs of highlighter pens at $3.10 per pack, twelve pencils at 8¢ each, and six 3-color pens at 79¢ each. How much money did he have left? (A) $4.17 (B) $5.00 (C) $6.57 (D) $7.12 (E) $8.00 3. 45 miles per hour is the same speed as _________________ inches per second. (A) 792 (B) 3,240 (C) 47,520 (D) 880 (E) 66 4. If P = {p,l,a,t,o}, O = {p,t,o,l,e,m,y}, and E = {e,u,c,l,i,d} then (P O) E = ? (A) { g } (B) {p,o,e} (C) {e, l } (D) {p,o,e,m} (E) {p,l,o,t} 5. An equation for the line shown is: (A) 3x qq 2y = 1 (B) 2x 3y = 2 (C) x y = q 1 (D) 3x 2y = qq 2 (E) x y = 1.5 6. Which of the following relations describes a function? (A) { (2,3), (3,3) (4,3) (5,3) } (B) { (qq 2,0), (0, 2) (0,2) (2,0) } (C) { (0,0), (2, q2) (2,2) (3,3) } (D) { (3,3), (3,4) (3,5) (3,6) } (E) none of these are a function 7. -
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. -
Newsletter 91
Newsletter 9 1: December 2010 Introduction This is the final nzmaths newsletter for 2010. It is also the 91 st we have produced for the website. You can have a look at some of the old newsletters on this page: http://nzmaths.co.nz/newsletter As you are no doubt aware, 91 is a very interesting and important number. A quick search on Wikipedia (http://en.wikipedia.org/wiki/91_%28number%29) will very quickly tell you that 91 is: • The atomic number of protactinium, an actinide. • The code for international direct dial phone calls to India • In cents of a U.S. dollar, the amount of money one has if one has one each of the coins of denominations less than a dollar (penny, nickel, dime, quarter and half dollar) • The ISBN Group Identifier for books published in Sweden. In more mathematically related trivia, 91 is: • the twenty-seventh distinct semiprime. • a triangular number and a hexagonal number, one of the few such numbers to also be a centered hexagonal number, and it is also a centered nonagonal number and a centered cube number. It is a square pyramidal number, being the sum of the squares of the first six integers. • the smallest positive integer expressible as a sum of two cubes in two different ways if negative roots are allowed (alternatively the sum of two cubes and the difference of two cubes): 91 = 6 3+(-5) 3 = 43+33. • the smallest positive integer expressible as a sum of six distinct squares: 91 = 1 2+2 2+3 2+4 2+5 2+6 2. -
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. -
Notations Used 1
NOTATIONS USED 1 NOTATIONS ⎡ (n −1)(m − 2)⎤ Tm,n = n 1+ - Gonal number of rank n with sides m . ⎣⎢ 2 ⎦⎥ n(n +1) T = - Triangular number of rank n . n 2 1 Pen = (3n2 − n) - Pentagonal number of rank n . n 2 2 Hexn = 2n − n - Hexagonal number of rank n . 1 Hep = (5n2 − 3n) - Heptagonal number of rank n . n 2 2 Octn = 3n − 2n - Octagonal number of rank n . 1 Nan = (7n2 − 5n) - Nanogonal number of rank n . n 2 2 Decn = 4n − 3n - Decagonal number of rank n . 1 HD = (9n 2 − 7n) - Hendecagonal number of rank n . n 2 1 2 DDn = (10n − 8n) - Dodecagonal number of rank n . 2 1 TD = (11n2 − 9n) - Tridecagonal number of rank n . n 2 1 TED = (12n 2 −10n) - Tetra decagonal number of rank n . n 2 1 PD = (13n2 −11n) - Pentadecagonal number of rank n . n 2 1 HXD = (14n2 −12n) - Hexadecagonal number of rank n . n 2 1 HPD = (15n2 −13n) - Heptadecagonal number of rank n . n 2 NOTATIONS USED 2 1 OD = (16n 2 −14n) - Octadecagonal number of rank n . n 2 1 ND = (17n 2 −15n) - Nonadecagonal number of rank n . n 2 1 IC = (18n 2 −16n) - Icosagonal number of rank n . n 2 1 ICH = (19n2 −17n) - Icosihenagonal number of rank n . n 2 1 ID = (20n 2 −18n) - Icosidigonal number of rank n . n 2 1 IT = (21n2 −19n) - Icositriogonal number of rank n . n 2 1 ICT = (22n2 − 20n) - Icositetragonal number of rank n . n 2 1 IP = (23n 2 − 21n) - Icosipentagonal number of rank n . -
The Schnirelmann Density of the Set of Deficient Numbers
THE SCHNIRELMANN DENSITY OF THE SET OF DEFICIENT NUMBERS A Thesis Presented to the Faculty of California State Polytechnic University, Pomona In Partial Fulfillment Of the Requirements for the Degree Master of Science In Mathematics By Peter Gerralld Banda 2015 SIGNATURE PAGE THESIS: THE SCHNIRELMANN DENSITY OF THE SET OF DEFICIENT NUMBERS AUTHOR: Peter Gerralld Banda DATE SUBMITTED: Summer 2015 Mathematics and Statistics Department Dr. Mitsuo Kobayashi Thesis Committee Chair Mathematics & Statistics Dr. Amber Rosin Mathematics & Statistics Dr. John Rock Mathematics & Statistics ii ACKNOWLEDGMENTS I would like to take a moment to express my sincerest appreciation of my fianc´ee’s never ceasing support without which I would not be here. Over the years our rela tionship has proven invaluable and I am sure that there will be many more fruitful years to come. I would like to thank my friends/co-workers/peers who shared the long nights, tears and triumphs that brought my mathematical understanding to what it is today. Without this, graduate school would have been lonely and I prob ably would have not pushed on. I would like to thank my teachers and mentors. Their commitment to teaching and passion for mathematics provided me with the incentive to work and succeed in this educational endeavour. Last but not least, I would like to thank my advisor and mentor, Dr. Mitsuo Kobayashi. Thanks to his patience and guidance, I made it this far with my sanity mostly intact. With his expertise in mathematics and programming, he was able to correct the direction of my efforts no matter how far they strayed. -
Arxiv:2106.08994V2 [Math.GM] 1 Aug 2021 Efc Ubr N30b.H Rvdta F2 If That and Proved Properties He Studied BC
Measuring Abundance with Abundancy Index Kalpok Guha∗ Presidency University, Kolkata Sourangshu Ghosh† Indian Institute of Technology Kharagpur, India Abstract A positive integer n is called perfect if σ(n) = 2n, where σ(n) denote n σ(n) the sum of divisors of . In this paper we study the ratio n . We de- I → I n σ(n) fine the function Abundancy Index : N Q with ( ) = n . Then we study different properties of Abundancy Index and discuss the set of Abundancy Index. Using this function we define a new class of num- bers known as superabundant numbers. Finally we study superabundant numbers and their connection with Riemann Hypothesis. 1 Introduction Definition 1.1. A positive integer n is called perfect if σ(n)=2n, where σ(n) denote the sum of divisors of n. The first few perfect numbers are 6, 28, 496, 8128, ... (OEIS A000396), This is a well studied topic in number theory. Euclid studied properties and nature of perfect numbers in 300 BC. He proved that if 2p −1 is a prime, then 2p−1(2p −1) is an even perfect number(Elements, Prop. IX.36). Later mathematicians have arXiv:2106.08994v2 [math.GM] 1 Aug 2021 spent years to study the properties of perfect numbers. But still many questions about perfect numbers remain unsolved. Two famous conjectures related to perfect numbers are 1. There exist infinitely many perfect numbers. Euler [1] proved that a num- ber is an even perfect numbers iff it can be written as 2p−1(2p − 1) and 2p − 1 is also a prime number.