DOCSLIB.ORG

Bi-Unitary Multiperfect Numbers, I

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
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. We aim to publish a series of papers on bi-unitary multiperfect numbers focusing on multiperfect numbers of the form n = 2au, where u is odd. In this paper—part I of the series—we investigate bi-unitary triperfect numbers of the form n = 2au, where 1 ≤ a ≤ 3. It appears that n = 120 = 2315 is the only such number. Hagis [6] found by computer the bi-unitary multiperfect numbers less than 107. We have found 31 such numbers up to 8:1010: The first 13 are due to Hagis. After complet- ing this paper we noticed that further numbers are already listed in The On-Line Encyclopedia of Integer Sequences (sequence A189000 Bi-unitary multiperfect numbers). The numbers listed there have been found by direct computer calculations. Our purpose is to present a mathematical search and treatment of bi-unitary multiperfect numbers. Keywords: Perfect numbers, Triperfect numbers, Multiperfect numbers, Bi-unitary analogues. 2010 Mathematics Subject Classification: 11A25. 93 1 Introduction Throughout this paper, all lower case letters denote positive integers; p and q denote primes. The letters u, v and w are reserved for odd numbers. Let σ(n) denote the sum of the positive divisors of n. It is well known that a positive integer n is called a perfect number if σ(n) = 2n; a concept introduced by Euclid (300 BC). He demonstrated that numbers of the form n = 2p−1(2p − 1) are perfect numbers, where p and 2p − 1 are primes. The converse that these numbers are the only even perfect numbers was established by Euler two thousand years later. Primes of the form 2p − 1, where p is a prime number, are called Mersenne primes and are denoted by Mp: Each Mersenne prime gives rise to an even perfect number. The first few examples of even perfect numbers are 6; 28; 496; 8128 and 33550336 which correspond to the primes p = 2; 3; 5; 7 and 13. One does not know whether this list of even perfect numbers can be continued indefinitely as the answer depends on whether there exist infinitely many Mersenne primes. Till now, 51 Mersenne primes are known (consequently 51 even perfect numbers). The internet information says that the 51-st Mersenne prime namely 282;589;933 − 1 was found by Patrick Laroche on December 7, 2018. This appears to be the largest known prime with 24,862,048 digits. Till to date, we do not know an example of an odd perfect number. Euler proved that if n is an odd perfect number, then n = pak2, where p is a prime not dividing the positive integer k and p ≡ a ≡ 1 (mod 4). In 2012, Pascal Ochem and Michael Rao (cf. [7]) proved that there are no odd perfect numbers less than 101500 and an odd perfect number should have at least 101 distinct prime factors. These appear to be the latest results of this type on odd perfect numbers. A positive integer n is called multiply perfect if σ(n) = kn for some positive integer k and the integer k is called the multiplicity of the number n; or equivalently n is multiply perfect if and only if njσ(n): Multiply perfect numbers with multiplicity k are called multi k-perfect numbers or k-fold perfect numbers. Multiply perfect numbers with multiplicity 2 are of course perfect numbers. Multi 3-perfect numbers are called triperfect numbers. For example, the smallest positive integers that are solutions of σ(n) = kn for k = 1; 2; 3; 4; 5 are, respectively, 1; 6; 120; 30240 and 14182439040 = 27:34:5:7:11217:19. Examples of multi k-perfect numbers are known for 1 ≤ k ≤ 11 (see Wikipedia). The number n = 2468:3140:566:1140:1711:1912 :::; a multiply perfect number of multiplicity 11 and with 1907 digits appears to have been found by G.F. Woltman on March 13, 2001. Till now, the only known odd multiply perfect number is 1. It is yet to be known whether there are finitely many multiply perfect numbers for each fixed multiplicity > 2 (see Wikipedia). An open question is whether each multi k-perfect number is divisible by k!: Note that L. E. Dick- son’s book [2] contains a good account of work on multiply perfect numbers (see also [3, 8]). A divisor d of n is called a unitary divisor if gcd(d; n=d) = 1. If d is a unitary divisor of n, we write dkn: The concept of unitary divisor was originally due to R. Vaidyanathaswamy [13] who called such a divisor as block factor. The present terminology is due to E. Cohen [1]. Let σ∗(n) denote the sum of the unitary divisors of n. It is known (cf. [1]) that σ∗ is a multiplicative function, σ∗(1) = 1 and σ∗(pa) = pa + 1 whenever p is a prime and a is a positive 94 integer. M. V. Subbarao and L. J. Warren [11] introduced the notion of a unitary perfect number as follows: a positive integer n is called a unitary perfect number if σ∗(n) = 2n: They proved that there are no odd unitary perfect numbers. The only five unitary perfect numbers known till now are 6; 60; 90; 87360 and 146361946186458562560000 = 218:3:54:7:11:13:19:37:79:109:157:313: The first four examples were due to M. V. Subbarao and L. J. Warren [11] and the last being due to C. R. Wall [14, 16]. It is not known whether there exists a unitary perfect number not divisible by 3 (see M. V. Subbarao and L. J. Warren [11]) and also whether there are infinitely many unitary perfect numbers (see M. V. Subbarao [10]). A positive integer m is called unitary multiply perfect if mjσ∗(m); that is, it is solution of σ∗(m) = km for some positive integer k. In the case k > 2, no example of a unitary multiply perfect number is known till now. It is known that if such a number exists, it must be even and greater than 10102 and must have more than forty four odd prime factors (see [4,5]). This problem is probably very difficult to settle. It is easily seen that if p is a prime, a and n are positive integers such that pajσ∗(n) and njσ∗(pa); (1:1) then m = pan (1:2) is unitary multiply perfect number. However it has been proved in [9] that the only solutions of (1:1) are p = 2; a = 1; n = 3; p = 3; a = 1; n = 2; and p = 3; a = 2; n = 10: The only solutions of (1:1) in the form (1:2) are m = 6 and m = 90; which are unitary perfect numbers. Thus if k > 3, there are no solutions to the equation σ∗(m) = km of the form (1:2). A divisor d of n is called a bi-unitary divisor if (d; n=d)∗∗ = 1; where the symbol (a; b)∗∗ denotes the greatest common unitary divisor of a and b. The concept of a bi-unitary divisor is due to D. Suryanarayana [12]. Let σ∗∗(n) denote the sum of bi-unitary divisors of n. The function σ∗∗(n) is multiplicative, that is, σ∗∗(1) = 1 and σ∗∗(mn) = σ∗∗(m)σ∗∗(n) whenever (m; n) = 1. If pα is a prime power and α is odd then every divisor of pα is a bi-unitary divisor; if α is even, each divisor of pα is a bi-unitary divisor except for pα/2: Hence ( α+1 σ(pα) = p −1 if α is odd; σ∗∗(pα) = p−1 (1:3) σ(pα) − pα/2 if α is even: If α is even, say α = 2k; then σ∗∗(pα) can be simplified to pk − 1 σ∗∗(pα) = :(pk+1 + 1): (1:4) p − 1 From (1.3), it is not difficult to observe that σ∗∗(n) is odd only when n = 1 or n = 2α: The concept of a bi-unitary perfect number was introduced by C.
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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 • , , •• , , , .
    [Show full text]
  • 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
    [Show full text]
  • 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.
    [Show full text]
  • Universitatis Scientiarum Budapestinensis De Rolando Eötvös Nominatae Sectio Computatorica
    ANNALES Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae sectio computatorica TOMUS XXXIV. REDIGIT I. KÁTAI ADIUVANTIBUS N.L. BASSILY, A. BENCZÚR, BUI MINH PHONG, Z. DARÓCZY, J. DEMETROVICS, R. FARZAN, S. FRIDLI, J. GALAMBOS, J. GONDA, Z. HORVÁTH, K.-H. INDLEKOFER, A. IVÁNYI, A. JÁRAI, J.-M. DE KONINCK, A. KÓSA, M. KOVÁCS, L. KOZMA, L. LAKATOS, P. RACSKÓ, F. SCHIPP, P. SIMON, G. STOYAN, L. SZILI, P.D. VARBANETS, L. VARGA, F. WEISZ 2011 Jarai´ Antal RESULTS ON CLASSES OF FUNCTIONAL EQUATIONS TRIBUTE TO ANTAL JARAI´ by J´anos Acz´el and Che Tat Ng While individual noncomposite functional equations in several variables had been solved at least since d’Alembert 1747 [9] and Cauchy 1821 [8], results on broad classes of such equations began appearing in the 1950’s and 1960’s. On general methods of solution see e.g. Acz´el [1] and for uniqueness of solutions Acz´el [2, 3], Acz´el and Hossz´u [6], Miller [20], Ng [21, 22], followed by several others. – Opening up and cultivating the field of regularization is mainly J´arai’s achievement. By regularization we mean assuming weaker regularity condi- tions, say measurability, of the unknown function and proving differentiability of several orders, for whole classes of functional equations. Differentiability of the unknown function(s) in the functional equation often leads to differential equations that are easier to solve. For example, in Acz´el and Chung [5] it was shown that locally Lebesgue integrable solutions of the functional equation n m fi(x + λiy)= pk(x)qk(y) i=1 k=1 holding for x, y on open real intervals, with appropriate independence between the functions, are in fact differentiable infinitely many times.
    [Show full text]
  • Exponential and Infinitary Divisors
    EXPONENTIAL AND INFINITARY DIVISORS ANDREW V. LELECHENKO Abstract. Our paper is devoted to several problems from the field of modified divisors: namely exponential and infinitary divisors. We study the behaviour of modified divisors, sum-of-divisors and totient functions. Main results con- cern with the asymptotic behaviour of mean values and explicit estimates of extremal orders. 1. Introduction Let m be an exponential divisor (or e-divisor) of n (denote m |(e) n) if m | n and for each prime p | n we have a | b, where pa || m, pb || n. This concept, introduced (e) by Subbarao [18], leads us to the e-divisor function τ (n)= m|(e)n 1 (sequence (e) A049419 in OEIS [20]) and sum-of-e-divisors function σ (n) = (e) m (se- P m| n quence A051377). These functions were studied by many authors, including among P others Wu and P´etermann [17, 25]. Consider a set of arithmetic functions A, a set of multiplicative prime-indepen- dent functions MP I and an operator E : A→MP I such that (Ef)(pa)= f(a). One can check that τ (e) = Eτ, but σ(e) 6= Eσ. Section 3 is devoted to the latter new function Eσ. On contrary several authors, including T´oth [22, 24] and P´etermann [16], studied exponential analogue of the totient function, defining φ(e) = Ef. However φ(e) lacks many significant properties of φ: it is prime-independent and φ(e) ≪ nε. In Section 4 we construct more natural modification of the totient function, which will be denoted by f(e).
    [Show full text]
  • Tables of Aliquot Cycles
    5/29/2017 Tables of Aliquot Cycles http://amicable.homepage.dk/tables.htm Go FEB MAY JUN f ❎ 117 captures 02 ⍰ ▾ About 28 May 2002 ‑ 2 May 2014 2012 2014 2015 this capture Tables of Aliquot Cycles Click on the number of known cycles to view the corresponding lists. Sociable Sociable Numbers Amicable Numbers Numbers Perfect Numbers of order four Type of other orders Known Last Known Last Known Last Known Last Cycles Update Cycles Update Cycles Update Cycles Update 28­Sep­ 01­Oct­ 13­Nov­ 11­Sep­ Ordinary 11994387 142 10 44 2007 2007 2006 2006 28­Sep­ 20­Nov­ 14­Jun­ 14­Dec­ Unitary 4911908 191 24 5 2007 2006 2005 1997 28­Sep­ 28­Sep­ 20­Nov­ 28­Sep­ Infinitary 11538100 5034 129 190 2007 2007 2006 2007 Exponential (note 3089296 28­Sep­ 371 20­Nov­ 38 15­Jun­ 12 06­Dec­ 2) 2007 2006 2005 1998 05­Feb­ 08­May­ 18­Oct­ 11­Mar­ Augmented 1931 2 0 note 1 2002 2003 1997 1998 05­Feb­ 06­Dec­ 06­Dec­ Augmented Unitary 27 0 0 2002 1998 1998 Augmented 10­Sep­ 06­Dec­ 06­Dec­ 425 0 0 Infinitary 2003 1998 1998 15­Feb­ 18­Oct­ 18­Oct­ 11­Mar­ Reduced 1946 0 1 0 2003 1997 1997 1998 Reduced Unitary 28 15­Feb­ 0 06­Dec­ 0 06­Dec­ http://web.archive.org/web/20140502102524/http://amicable.homepage.dk/tables.htm 1/3 5/29/2017 Tables of Aliquot Cycles http://amicable.homepage.dk/tables.htm2003 1998 Go 1998 FEB MAY JUN f ❎ 117 captures 10­Sep­ 06­Dec­ 06­Dec­ 02 ⍰ Reduced Infinitary 427 0 0 ▾ About 28 May 2002 ‑ 2 May 2014 2003 1998 1998 2012 2014 2015 this capture Note 1: All powers of 2 are augmented perfect numbers; no other augmented perfect numbers are known.
    [Show full text]
  • New Results on Some Multiplicative Functions
    NNTDM 17 (2011), 2, 18-30 New results on some multiplicative functions Mladen Vassilev - Missana1, Peter Vassilev2 1 5 V. Hugo Str., Sofia–1124, Bulgaria e-mail:[email protected] 2 Institute of Biophysics and Biomedical Engineering e-mail:[email protected] Abstract The paper is a continuation of [1]. The considerations are over the class of mul- tiplicative functions with strictly positive values and more precisely, over the pairs (f; g) of such functions, which have a special property, called in the paper property S: For every such pair (f; g) and for every composite number n > 1; the problem of (︀ n )︀ finding the maximum and minimum of the numbers f(d)g d ; when d runs over all proper divisors of n; is completely solved. Since some classical multiplicative func- tions like Euler's totient function '; Dedekind's function ; the sum of all divisors of m; i.e. 휎(m); the number of all divisors of m; i.e. 휏(m); and 2!(m) (where !(m) is the number of all prime divisors of m) form pairs having property S; we apply our results to these functions and also resolve the questions of finding the maximum (︀ n )︀ (︀ n )︀ (︀ n )︀ !(d) (︀ n )︀ and minimum of the numbers '(d)휎 d ;'(d) d ; 휏(d)휎 d ; 2 휎 d ; where d runs over all proper divisors of n: In addition some corollaries from the obtained results, concerning unitary proper divisors, are made. Since many other pairs of multiplicative functions (except the considered in the paper) have property S; they may be investigated in similar manner in a future research.
    [Show full text]
  • Numbers 1 to 100
    Numbers 1 to 100 PDF generated using the open source mwlib toolkit. See http://code.pediapress.com/ for more information. PDF generated at: Tue, 30 Nov 2010 02:36:24 UTC Contents Articles −1 (number) 1 0 (number) 3 1 (number) 12 2 (number) 17 3 (number) 23 4 (number) 32 5 (number) 42 6 (number) 50 7 (number) 58 8 (number) 73 9 (number) 77 10 (number) 82 11 (number) 88 12 (number) 94 13 (number) 102 14 (number) 107 15 (number) 111 16 (number) 114 17 (number) 118 18 (number) 124 19 (number) 127 20 (number) 132 21 (number) 136 22 (number) 140 23 (number) 144 24 (number) 148 25 (number) 152 26 (number) 155 27 (number) 158 28 (number) 162 29 (number) 165 30 (number) 168 31 (number) 172 32 (number) 175 33 (number) 179 34 (number) 182 35 (number) 185 36 (number) 188 37 (number) 191 38 (number) 193 39 (number) 196 40 (number) 199 41 (number) 204 42 (number) 207 43 (number) 214 44 (number) 217 45 (number) 220 46 (number) 222 47 (number) 225 48 (number) 229 49 (number) 232 50 (number) 235 51 (number) 238 52 (number) 241 53 (number) 243 54 (number) 246 55 (number) 248 56 (number) 251 57 (number) 255 58 (number) 258 59 (number) 260 60 (number) 263 61 (number) 267 62 (number) 270 63 (number) 272 64 (number) 274 66 (number) 277 67 (number) 280 68 (number) 282 69 (number) 284 70 (number) 286 71 (number) 289 72 (number) 292 73 (number) 296 74 (number) 298 75 (number) 301 77 (number) 302 78 (number) 305 79 (number) 307 80 (number) 309 81 (number) 311 82 (number) 313 83 (number) 315 84 (number) 318 85 (number) 320 86 (number) 323 87 (number) 326 88 (number)
    [Show full text]