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On Untouchable Numbers and Related Problems
ON UNTOUCHABLE NUMBERS AND RELATED PROBLEMS CARL POMERANCE AND HEE-SUNG YANG Abstract. In 1973, Erd˝os proved that a positive proportion of numbers are untouchable, that is, not of the form σ(n) n, the sum of the proper divisors of n. We investigate the analogous question where σ is− replaced with the sum-of-unitary-divisors function σ∗ (which sums divisors d of n such that (d, n/d) = 1), thus solving a problem of te Riele from 1976. We also describe a fast algorithm for enumerating untouchable numbers and their kin. 1. Introduction If f(n) is an arithmetic function with nonnegative integral values it is interesting to con- sider Vf (x), the number of integers 0 m x for which f(n) = m has a solution. That is, one might consider the distribution≤ of the≤ range of f within the nonnegative integers. For some functions f(n) this is easy, such as the function f(n) = n, where Vf (x) = x , or 2 ⌊ ⌋ f(n) = n , where Vf (x) = √x . For f(n) = ϕ(n), Euler’s ϕ-function, it was proved by ⌊ ⌋ 1+o(1) Erd˝os [Erd35] in 1935 that Vϕ(x) = x/(log x) as x . Actually, the same is true for a number of multiplicative functions f, such as f = σ,→ the ∞ sum-of-divisors function, and f = σ∗, the sum-of-unitary-divisors function, where we say d is a unitary divisor of n if d n, | d > 0, and gcd(d,n/d) = 1. In fact, a more precise estimation of Vf (x) is known in these cases; see Ford [For98]. -
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. -
Anti-Conformism and Social Networks Alexis Poindron
Anti-conformism and Social Networks Alexis Poindron To cite this version: Alexis Poindron. Anti-conformism and Social Networks. Economics and Finance. 2016. dumas- 02102411 HAL Id: dumas-02102411 https://dumas.ccsd.cnrs.fr/dumas-02102411 Submitted on 18 Apr 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Anti-conformism and Social Networks Alexis Poindron Université Paris I Sorbonne, UFR de sciences économiques 02 Master 2 Économie Théorique et Empirique 2015-2016 Mémoire présenté et soutenu par Alexis Poindron le 17/05/2016 Sous la direction de Michel Grabisch et Agnieszka Rusinowska May 17, 2016 L’Université de paris I Panthéon Sorbonne n’entend donner aucune approbation, ni désappro- bation aux opinions émises dans ce mémoire ; elle doivent être considérées comme propres à leur auteur. 1 Contents 1 Introduction . .3 1.1 Literature review and context of this master thesis . .3 1.2 Some notations and definitions . .5 2 Generalized OWA . .6 2.1 GOWA: definitions and exemples . .6 2.2 Discussion on GOWA . .8 3 Terminal states and classes with GOWA: homogeneous framework . .9 3.1 Preliminary results . .9 3.2 List of possible terminal classes and unicity . -
Rapid Arithmetic
RAPID ARITHMETIC QUICK AND SPE CIAL METHO DS IN ARITH METICAL CALCULATION TO GETHE R WITH A CO LLE CTION OF PUZZLES AND CURI OSITIES OF NUMBERS ’ D D . T . O CONOR SLOAN E , PH . LL . “ ’ " “ A or o Anda o Ele ri i ta da rd le i a Di ion uth f mfic f ct c ty, S n E ctr c l ct " a Eleme a a i s r n r Elec r a l C lcul on etc . etc y, tp y t ic t , , . NE W Y ORK D VA N NOS D O M N . TRAN C PA Y E mu? WAuw Sum 5 PRINTED IN THE UNITED STATES OF AMERICA which receive little c n in ne or but s a t treatment text books . If o o o f doin i e meth d g an operat on is giv n , it is considered n e ough. But it is certainly interesting to know that there e o f a t are a doz n or more methods adding, th there are a of number of ways applying the other three primary rules , and to find that it is quite within the reach of anyo ne to io add up two columns simultaneously . The multiplicat n table for some reason stops abruptly at twelve times ; it is not to a on or hard c rry it to at least towards twenty times . n e n o too Taking up the questio of xpone ts , it is not g ing far to assert that many college graduates do not under i a nd stand the meaning o a fractional exponent , as few can tell why any number great or small raised to the zero power is equa l to one when it seems a s if it ought to be equal to zero . -
Variant of a Theorem of Erdős on The
MATHEMATICS OF COMPUTATION Volume 83, Number 288, July 2014, Pages 1903–1913 S 0025-5718(2013)02775-5 Article electronically published on October 29, 2013 VARIANT OF A THEOREM OF ERDOS˝ ON THE SUM-OF-PROPER-DIVISORS FUNCTION CARL POMERANCE AND HEE-SUNG YANG Abstract. In 1973, Erd˝os proved that a positive proportion of numbers are not of the form σ(n) − n, the sum of the proper divisors of n.Weprovethe analogous result where σ is replaced with the sum-of-unitary-divisors function σ∗ (which sums divisors d of n such that (d, n/d) = 1), thus solving a problem of te Riele from 1976. We also describe a fast algorithm for enumerating numbers not in the form σ(n) − n, σ∗(n) − n,andn − ϕ(n), where ϕ is Euler’s function. 1. Introduction If f(n) is an arithmetic function with nonnegative integral values it is interesting to consider Vf (x), the number of integers 0 ≤ m ≤ x for which f(n)=m has a solution. That is, one might consider the distribution of the image of f within the nonnegative integers. For some functions f(n) this is easy, such√ as the function 2 f(n)=n,whereVf (x)=x,orf(n)=n ,whereVf (x)= x.Forf(n)= ϕ(n), Euler’s ϕ-function, it was proved by Erd˝os [Erd35] in 1935 that Vϕ(x)= x/(log x)1+o(1) as x →∞. Actually, the same is true for a number of multiplicative functions f,suchasf = σ, the sum-of-divisors function, and f = σ∗,thesum-of- unitary-divisors function, where we say d is a unitary divisor of n if d | n, d>0, and gcd(d, n/d) = 1. -
Social Networks IZA Dpno
IZA DP No. 4621 Social Networks Joan de Martí Yves Zenou December 2009 DISCUSSION PAPER SERIES Forschungsinstitut zur Zukunft der Arbeit Institute for the Study of Labor Social Networks Joan de Martí Universitat Pompeu Fabra and Barcelona GSE Yves Zenou Stockholm University, IFN and IZA Discussion Paper No. 4621 December 2009 IZA P.O. Box 7240 53072 Bonn Germany Phone: +49-228-3894-0 Fax: +49-228-3894-180 E-mail: [email protected] Any opinions expressed here are those of the author(s) and not those of IZA. Research published in this series may include views on policy, but the institute itself takes no institutional policy positions. The Institute for the Study of Labor (IZA) in Bonn is a local and virtual international research center and a place of communication between science, politics and business. IZA is an independent nonprofit organization supported by Deutsche Post Foundation. The center is associated with the University of Bonn and offers a stimulating research environment through its international network, workshops and conferences, data service, project support, research visits and doctoral program. IZA engages in (i) original and internationally competitive research in all fields of labor economics, (ii) development of policy concepts, and (iii) dissemination of research results and concepts to the interested public. IZA Discussion Papers often represent preliminary work and are circulated to encourage discussion. Citation of such a paper should account for its provisional character. A revised version may be available directly from the author. IZA Discussion Paper No. 4621 December 2009 ABSTRACT Social Networks We survey the literature on social networks by putting together the economics, sociological and physics/applied mathematics approaches, showing their similarities and differences. -
The Mathematical Sociologist
The Mathematical Sociologist Newsletter of the Mathematical Sociology Section of the American Sociological Association Spring 2003 Section Officers Chair From the newsletter editor Noah E. Friedkin, Barbara Meeker, [email protected] University of California, Santa Barbara The MathSoc Section Newsletter is [email protected] back by popular demand (sort of). After a gap of Chair Elect: about two years, I volunteered once again to David Heise, Indiana University produce the newsletter. Given the many very [email protected] competitive professional accomplishments we all Past Chair, and: Section Nominations try for, it was something of a relief to campaign Committee Chair for a job absolutely no one else wants! I intend to Patrick Doreian, University of Pittsburgh do either one or two issues a year. This is the [email protected] issue for 2002-2003, covering approximately Secretary/Treasurer – 2005 April 2002 through April 2003. Dave McFarland Lisa Troyer, University of Iowa (UCLA - [email protected]) [email protected] has agreed to post newsletters and other information on a Section web page. Council 2003 The Section has been very active, as Kenneth C. Land, Duke University shown in the minutes of the Business Meeting, [email protected] prepared by 2002 Secretary-Treasurer Joe Whitmeyer, (see below). Other activities of the Jane Sell, Texas A&M University Section during the past year and some with ASA [email protected] coming up are also described below. Section Chair Noah Friedkin has written Council - 2004 an interesting comment on the Current State of Diane H. Felmlee, Mathematical Sociology; I think most of us University of California-Davis would agree wholeheartedly with his remarks. -
The Project Gutenberg Ebook #40624: a Scrap-Book Of
The Project Gutenberg EBook of A Scrap-Book of Elementary Mathematics, by William F. White This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.org Title: A Scrap-Book of Elementary Mathematics Notes, Recreations, Essays Author: William F. White Release Date: August 30, 2012 [EBook #40624] Language: English Character set encoding: ISO-8859-1 *** START OF THIS PROJECT GUTENBERG EBOOK A SCRAP-BOOK *** Produced by Andrew D. Hwang, Joshua Hutchinson, and the Online Distributed Proofreading Team at http://www.pgdp.net (This file was produced from images from the Cornell University Library: Historical Mathematics Monographs collection.) Transcriber’s Note Minor typographical corrections, presentational changes, and no- tational modernizations have been made without comment. In- ternal references to page numbers may be off by one. All changes are detailed in the LATEX source file, which may be downloaded from www.gutenberg.org/ebooks/40624. This PDF file is optimized for screen viewing, but may easily be recompiled for printing. Please consult the preamble of the LATEX source file for instructions. NUMERALS OR COUNTERS? From the Margarita Philosophica. (See page 48.) A Scrap-Book of Elementary Mathematics Notes, Recreations, Essays By William F. White, Ph.D. State Normal School, New Paltz, New York Chicago The Open Court Publishing Company London Agents Kegan Paul, Trench, Trübner & Co., Ltd. 1908 Copyright by The Open Court Publishing Co. -
Abbreviations of Names of Serials
Abbreviations of Names of Serials This list gives the form of references used in Mathematical Reviews (MR). ∗ not previously listed The abbreviation is followed by the complete title, the place of publication x journal indexed cover-to-cover and other pertinent information. y monographic series Update date: January 30, 2018 4OR 4OR. A Quarterly Journal of Operations Research. Springer, Berlin. ISSN xActa Math. Appl. Sin. Engl. Ser. Acta Mathematicae Applicatae Sinica. English 1619-4500. Series. Springer, Heidelberg. ISSN 0168-9673. y 30o Col´oq.Bras. Mat. 30o Col´oquioBrasileiro de Matem´atica. [30th Brazilian xActa Math. Hungar. Acta Mathematica Hungarica. Akad. Kiad´o,Budapest. Mathematics Colloquium] Inst. Nac. Mat. Pura Apl. (IMPA), Rio de Janeiro. ISSN 0236-5294. y Aastaraam. Eesti Mat. Selts Aastaraamat. Eesti Matemaatika Selts. [Annual. xActa Math. Sci. Ser. A Chin. Ed. Acta Mathematica Scientia. Series A. Shuxue Estonian Mathematical Society] Eesti Mat. Selts, Tartu. ISSN 1406-4316. Wuli Xuebao. Chinese Edition. Kexue Chubanshe (Science Press), Beijing. ISSN y Abel Symp. Abel Symposia. Springer, Heidelberg. ISSN 2193-2808. 1003-3998. y Abh. Akad. Wiss. G¨ottingenNeue Folge Abhandlungen der Akademie der xActa Math. Sci. Ser. B Engl. Ed. Acta Mathematica Scientia. Series B. English Wissenschaften zu G¨ottingen.Neue Folge. [Papers of the Academy of Sciences Edition. Sci. Press Beijing, Beijing. ISSN 0252-9602. in G¨ottingen.New Series] De Gruyter/Akademie Forschung, Berlin. ISSN 0930- xActa Math. Sin. (Engl. Ser.) Acta Mathematica Sinica (English Series). 4304. Springer, Berlin. ISSN 1439-8516. y Abh. Akad. Wiss. Hamburg Abhandlungen der Akademie der Wissenschaften xActa Math. Sinica (Chin. Ser.) Acta Mathematica Sinica. -
Jacob Gates Foster
JACOB GATES FOSTER Department of Sociology, UCLA 264 Haines Hall, 375 Portola Plaza Los Angeles, CA 90095 Email: [email protected] Website: http://www.sociology.ucla.edu/faculty/jacob-foster Phone: 1-312-608-8742 EMPLOYMENT 2013— Assistant Professor, Department of Sociology, UCLA Affiliate, California Center for Population Research Affiliate, Center for Social Statistics Affiliate, BRITE Center Core Faculty, Center for Behavior, Evolution, and Culture Executive Committee, Institute for Digital Research and Education 2013 Research Associate (Assistant Professor), Department of Sociology, University of Chicago 2010-2012 Postdoctoral Scholar, Department of Sociology, University of Chicago supervised by James A. Evans EDUCATION 2006-2010 PhD, Physics, University of Calgary Thesis: “On the Methodology of Complex Network Analysis,” supervised by Profs. Maya Paczuski and Peter Grassberger 2003-2006 DPhil Candidate, Mathematics, University of Oxford Transfer Thesis: “Conformal Invariance, Renormalization, and the Eternal Universe,” supervised by Prof. Sir Roger Penrose 1999-2003 B.S. with distinction, magna cum laude, Physics, Duke University Honors Thesis: “Physics with Two Time Dimensions,” supervised by Prof. Berndt Mueller RESEARCH INTERESTS Science and technology, social theory, computational social science, networks, complex systems, cognition and culture, machine learning, cultural evolution, game theory FUNDING 2018-2021 PI, “Building a Community of Practice in Diverse Intelligences,” Templeton World Charity Foundation 2018-2019 PI, -
CASAS Math Standards
CASAS Math Standards M1 Number Sense M2 Algebra M3 Geometry M4 Measurement M5 Statistics, Data Analysis and Probability NRS LEVELS FOR ABE/ASE 1 2 3 4 5 6 CASAS LEVELS A B B C D E M1 Number sense M1.1 Read, write, order and compare rational numbers M1.1.1 Associate numbers with quantities M1.1.2 Count with whole numbers M1.1.3 Count by 2s, 5s, and 10s up to 100 M1.1.4 Recognize odd and even numbers M1.1.5 Understand the decimal place value system: read, write, order and compare whole and decimal numbers (e.g., 0.13 > 0.013 because 13/100 > 13/1000) M1.1.6 Round off numbers to the nearest 10, 100, 1000 and/or to the nearest whole number, tenth, hundredth or thousandth according to the demands of the context M1.1.7 Using place value, compose and decompose numbers with up to 5 digits and/or with three decimal places (e.g., 54.8 = 5 × 10 + 4 × 1 + 8 × 0.1) M1.1.8 Interpret and use a fraction in context (e.g., as a portion of a whole area or set) M1.1.9 Find equivalent fractions and simplify fractions to lowest terms M1.1.10 Use common fractions to estimate the relationship between two quantities (e.g., 31/179 is close to 1/6) M1.1.11 Convert between mixed numbers and improper fractions M1.1.12 Use common fractions and their decimal equivalents interchangeably M1.1.13 Read, write, order and compare positive and negative real numbers (integers, decimals, and fractions) M1.1.14 Interpret and use scientific notation M1.2 Demonstrate understanding of the operations of addition and subtraction, their relation to each -
John Levi Martin Department of Sociology 773/702-7098 University
1 John Levi Martin Department of Sociology 773/702-7098 University of Chicago [email protected] 1126 East 59th Street http://home.uchicago.edu/~jlmartin/ Chicago, Illinois 60637 EDUCATION AND EMPLOYMENT Positions: 2013- Florence Borchert Bartling Professor of Sociology, University of Chicago. Faculty Award for Excellence in Graduate Teaching and Mentoring, 2015 2009-2013 Professor of Sociology, University of Chicago. 2008-2009 Visiting Professor of Sociology, University of Chicago. 2007-2009 Professor of Sociology, University of California, Berkeley. 2007-2009 Romnes Research Fellow, University of Wisconsin, Madison. 2006-2009 Professor of Sociology, University of Wisconsin, Madison. 2003-2006 Associate Professor of Sociology, University of Wisconsin, Madison. 2003-2005 Associate Professor of Sociology, Rutgers University, New Brunswick (on leave). 1997-2003 Assistant Professor of Sociology, Rutgers University, New Brunswick. Education: 1997 PhD. Sociology, University of California, Berkeley. Committee: Ann Swidler, Mike Hout, James Wiley, John Wilmoth. Dissertation: Power Structure and Belief Structure in Forty American Communes. 1990 MA. Sociology, University of California, Berkeley. Methods paper: “The Use of Loglinear Techniques in the Analysis of Indicator Structure.” 1987 BA with high honors in Sociology and English, Wesleyan University. Thesis: The Epistemology of Fundamentalism. Herbert Hyman prize for undergraduate sociology thesis Roura-Parella prize for “catholic curiosity and general learning” Phi Beta Kappa 2 WORKS Books: In The True, the Good and the Beautiful: On the Rise and Fall of the Preparation Kantian Grammar of Action. In Thinking Through Statistics. Preparation In Press Thinking Through Methods. University of Chicago Press. 2015 Thinking Through Theory. Norton. 2011 The Explanation of Social Action. Oxford University Press.