Some Arithmetic Functions of Factorials in Lucas Sequences
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Catalan Numbers Modulo 2K
1 2 Journal of Integer Sequences, Vol. 13 (2010), 3 Article 10.5.4 47 6 23 11 Catalan Numbers Modulo 2k Shu-Chung Liu1 Department of Applied Mathematics National Hsinchu University of Education Hsinchu, Taiwan [email protected] and Jean C.-C. Yeh Department of Mathematics Texas A & M University College Station, TX 77843-3368 USA Abstract In this paper, we develop a systematic tool to calculate the congruences of some combinatorial numbers involving n!. Using this tool, we re-prove Kummer’s and Lucas’ theorems in a unique concept, and classify the congruences of the Catalan numbers cn (mod 64). To achieve the second goal, cn (mod 8) and cn (mod 16) are also classified. Through the approach of these three congruence problems, we develop several general properties. For instance, a general formula with powers of 2 and 5 can evaluate cn (mod k 2 ) for any k. An equivalence cn ≡2k cn¯ is derived, wheren ¯ is the number obtained by partially truncating some runs of 1 and runs of 0 in the binary string [n]2. By this equivalence relation, we show that not every number in [0, 2k − 1] turns out to be a k residue of cn (mod 2 ) for k ≥ 2. 1 Introduction Throughout this paper, p is a prime number and k is a positive integer. We are interested in enumerating the congruences of various combinatorial numbers modulo a prime power 1Partially supported by NSC96-2115-M-134-003-MY2 1 k q := p , and one of the goals of this paper is to classify Catalan numbers cn modulo 64. -
On Fixed Points of Iterations Between the Order of Appearance and the Euler Totient Function
mathematics Article On Fixed Points of Iterations Between the Order of Appearance and the Euler Totient Function ŠtˇepánHubálovský 1,* and Eva Trojovská 2 1 Department of Applied Cybernetics, Faculty of Science, University of Hradec Králové, 50003 Hradec Králové, Czech Republic 2 Department of Mathematics, Faculty of Science, University of Hradec Králové, 50003 Hradec Králové, Czech Republic; [email protected] * Correspondence: [email protected] or [email protected]; Tel.: +420-49-333-2704 Received: 3 October 2020; Accepted: 14 October 2020; Published: 16 October 2020 Abstract: Let Fn be the nth Fibonacci number. The order of appearance z(n) of a natural number n is defined as the smallest positive integer k such that Fk ≡ 0 (mod n). In this paper, we shall find all positive solutions of the Diophantine equation z(j(n)) = n, where j is the Euler totient function. Keywords: Fibonacci numbers; order of appearance; Euler totient function; fixed points; Diophantine equations MSC: 11B39; 11DXX 1. Introduction Let (Fn)n≥0 be the sequence of Fibonacci numbers which is defined by 2nd order recurrence Fn+2 = Fn+1 + Fn, with initial conditions Fi = i, for i 2 f0, 1g. These numbers (together with the sequence of prime numbers) form a very important sequence in mathematics (mainly because its unexpectedly and often appearance in many branches of mathematics as well as in another disciplines). We refer the reader to [1–3] and their very extensive bibliography. We recall that an arithmetic function is any function f : Z>0 ! C (i.e., a complex-valued function which is defined for all positive integer). -
Algebra & Number Theory Vol. 7 (2013)
Algebra & Number Theory Volume 7 2013 No. 3 msp Algebra & Number Theory msp.org/ant EDITORS MANAGING EDITOR EDITORIAL BOARD CHAIR Bjorn Poonen David Eisenbud Massachusetts Institute of Technology University of California Cambridge, USA Berkeley, USA BOARD OF EDITORS Georgia Benkart University of Wisconsin, Madison, USA Susan Montgomery University of Southern California, USA Dave Benson University of Aberdeen, Scotland Shigefumi Mori RIMS, Kyoto University, Japan Richard E. Borcherds University of California, Berkeley, USA Raman Parimala Emory University, USA John H. Coates University of Cambridge, UK Jonathan Pila University of Oxford, UK J-L. Colliot-Thélène CNRS, Université Paris-Sud, France Victor Reiner University of Minnesota, USA Brian D. Conrad University of Michigan, USA Karl Rubin University of California, Irvine, USA Hélène Esnault Freie Universität Berlin, Germany Peter Sarnak Princeton University, USA Hubert Flenner Ruhr-Universität, Germany Joseph H. Silverman Brown University, USA Edward Frenkel University of California, Berkeley, USA Michael Singer North Carolina State University, USA Andrew Granville Université de Montréal, Canada Vasudevan Srinivas Tata Inst. of Fund. Research, India Joseph Gubeladze San Francisco State University, USA J. Toby Stafford University of Michigan, USA Ehud Hrushovski Hebrew University, Israel Bernd Sturmfels University of California, Berkeley, USA Craig Huneke University of Virginia, USA Richard Taylor Harvard University, USA Mikhail Kapranov Yale University, USA Ravi Vakil Stanford University, -
Arxiv:Math/0201082V1
2000]11A25, 13J05 THE RING OF ARITHMETICAL FUNCTIONS WITH UNITARY CONVOLUTION: DIVISORIAL AND TOPOLOGICAL PROPERTIES. JAN SNELLMAN Abstract. We study (A, +, ⊕), the ring of arithmetical functions with unitary convolution, giving an isomorphism between (A, +, ⊕) and a generalized power series ring on infinitely many variables, similar to the isomorphism of Cashwell- Everett[4] between the ring (A, +, ·) of arithmetical functions with Dirichlet convolution and the power series ring C[[x1,x2,x3,... ]] on countably many variables. We topologize it with respect to a natural norm, and shove that all ideals are quasi-finite. Some elementary results on factorization into atoms are obtained. We prove the existence of an abundance of non-associate regular non-units. 1. Introduction The ring of arithmetical functions with Dirichlet convolution, which we’ll denote by (A, +, ·), is the set of all functions N+ → C, where N+ denotes the positive integers. It is given the structure of a commutative C-algebra by component-wise addition and multiplication by scalars, and by the Dirichlet convolution f · g(k)= f(r)g(k/r). (1) Xr|k Then, the multiplicative unit is the function e1 with e1(1) = 1 and e1(k) = 0 for k> 1, and the additive unit is the zero function 0. Cashwell-Everett [4] showed that (A, +, ·) is a UFD using the isomorphism (A, +, ·) ≃ C[[x1, x2, x3,... ]], (2) where each xi corresponds to the function which is 1 on the i’th prime number, and 0 otherwise. Schwab and Silberberg [9] topologised (A, +, ·) by means of the norm 1 arXiv:math/0201082v1 [math.AC] 10 Jan 2002 |f| = (3) min { k f(k) 6=0 } They noted that this norm is an ultra-metric, and that ((A, +, ·), |·|) is a valued ring, i.e. -
Asymptotics of Multivariate Sequences, Part III: Quadratic Points
Asymptotics of multivariate sequences, part III: quadratic points Yuliy Baryshnikov 1 Robin Pemantle 2,3 ABSTRACT: We consider a number of combinatorial problems in which rational generating func- tions may be obtained, whose denominators have factors with certain singularities. Specifically, there exist points near which one of the factors is asymptotic to a nondegenerate quadratic. We compute the asymptotics of the coefficients of such a generating function. The computation requires some topological deformations as well as Fourier-Laplace transforms of generalized functions. We apply the results of the theory to specific combinatorial problems, such as Aztec diamond tilings, cube groves, and multi-set permutations. Keywords: generalized function, Fourier transform, Fourier-Laplace, lacuna, multivariate generating function, hyperbolic polynomial, amoeba, Aztec diamond, quantum random walk, random tiling, cube grove. Subject classification: Primary: 05A16 ; Secondary: 83B20, 35L99. 1Bell Laboratories, Lucent Technologies, 700 Mountain Avenue, Murray Hill, NJ 07974-0636, [email protected] labs.com 2Research supported in part by National Science Foundation grant # DMS 0603821 3University of Pennsylvania, Department of Mathematics, 209 S. 33rd Street, Philadelphia, PA 19104 USA, pe- [email protected] Contents 1 Introduction 1 1.1 Background and motivation . 1 1.2 Methods and organization . 4 1.3 Comparison with other techniques . 7 2 Notation and preliminaries 8 2.1 The Log map and amoebas . 9 2.2 Dual cones, tangent cones and normal cones . 10 2.3 Hyperbolicity for homogeneous polynomials . 11 2.4 Hyperbolicity and semi-continuity for log-Laurent polynomials on the amoeba boundary 14 2.5 Critical points . 20 2.6 Quadratic forms and their duals . -
Fixed Points of Certain Arithmetic Functions
FIXED POINTS OF CERTAIN ARITHMETIC FUNCTIONS WALTER E. BECK and RUDOLPH M. WAJAR University of Wisconsin, Whitewater, Wisconsin 53190 IWTRODUCTIOW Perfect, amicable and sociable numbers are fixed points of the arithemetic function L and its iterates, L (n) = a(n) - n, where o is the sum of divisor's function. Recently there have been investigations into functions differing from L by 1; i.e., functions L+, Z.,, defined by L + (n)= L(n)± 1. Jerrard and Temperley [1] studied the existence of fixed points of L+ and /._. Lai and Forbes [2] conducted a computer search for fixed points of (LJ . For earlier references to /._, see the bibliography in [2]. We consider the analogous situation using o*, the sum of unitary divisors function. Let Z.J, Lt, be arithmetic functions defined by L*+(n) = o*(n)-n±1. In § 1, we prove, using parity arguments, that L* has no fixed points. Fixed points of iterates of L* arise in sets where the number of elements in the set is equal to the power of/.* in question. In each such set there is at least one natural number n such that L*(n) > n. In § 2, we consider conditions n mustsatisfy to enjoy the inequality and how the inequality acts under multiplication. In particular if n is even, it is divisible by at least three primes; if odd, by five. If /? enjoys the inequality, any multiply by a relatively prime factor does so. There is a bound on the highest power of n that satisfies the inequality. Further if n does not enjoy the inequality, there are bounds on the prime powers multiplying n which will yield the inequality. -
On Some New Arithmetic Functions Involving Prime Divisors and Perfect Powers
On some new arithmetic functions involving prime divisors and perfect powers. Item Type Article Authors Bagdasar, Ovidiu; Tatt, Ralph-Joseph Citation Bagdasar, O., and Tatt, R. (2018) ‘On some new arithmetic functions involving prime divisors and perfect powers’, Electronic Notes in Discrete Mathematics, 70, pp.9-15. doi: 10.1016/ j.endm.2018.11.002 DOI 10.1016/j.endm.2018.11.002 Publisher Elsevier Journal Electronic Notes in Discrete Mathematics Download date 30/09/2021 07:19:18 Item License http://creativecommons.org/licenses/by/4.0/ Link to Item http://hdl.handle.net/10545/623232 On some new arithmetic functions involving prime divisors and perfect powers Ovidiu Bagdasar and Ralph Tatt 1,2 Department of Electronics, Computing and Mathematics University of Derby Kedleston Road, Derby, DE22 1GB, United Kingdom Abstract Integer division and perfect powers play a central role in numerous mathematical results, especially in number theory. Classical examples involve perfect squares like in Pythagora’s theorem, or higher perfect powers as the conjectures of Fermat (solved in 1994 by A. Wiles [8]) or Catalan (solved in 2002 by P. Mih˘ailescu [4]). The purpose of this paper is two-fold. First, we present some new integer sequences a(n), counting the positive integers smaller than n, having a maximal prime factor. We introduce an arithmetic function counting the number of perfect powers ij obtained for 1 ≤ i, j ≤ n. Along with some properties of this function, we present the sequence A303748, which was recently added to the Online Encyclopedia of Integer Sequences (OEIS) [5]. Finally, we discuss some other novel integer sequences. -
Generalized Catalan Numbers and Some Divisibility Properties
UNLV Theses, Dissertations, Professional Papers, and Capstones December 2015 Generalized Catalan Numbers and Some Divisibility Properties Jacob Bobrowski University of Nevada, Las Vegas Follow this and additional works at: https://digitalscholarship.unlv.edu/thesesdissertations Part of the Mathematics Commons Repository Citation Bobrowski, Jacob, "Generalized Catalan Numbers and Some Divisibility Properties" (2015). UNLV Theses, Dissertations, Professional Papers, and Capstones. 2518. http://dx.doi.org/10.34917/8220086 This Thesis is protected by copyright and/or related rights. It has been brought to you by Digital Scholarship@UNLV with permission from the rights-holder(s). You are free to use this Thesis in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s) directly, unless additional rights are indicated by a Creative Commons license in the record and/ or on the work itself. This Thesis has been accepted for inclusion in UNLV Theses, Dissertations, Professional Papers, and Capstones by an authorized administrator of Digital Scholarship@UNLV. For more information, please contact [email protected]. GENERALIZED CATALAN NUMBERS AND SOME DIVISIBILITY PROPERTIES By Jacob Bobrowski Bachelor of Arts - Mathematics University of Nevada, Las Vegas 2013 A thesis submitted in partial fulfillment of the requirements for the Master of Science - Mathematical Sciences College of Sciences Department of Mathematical Sciences The Graduate College University of Nevada, Las Vegas December 2015 Thesis Approval The Graduate College The University of Nevada, Las Vegas November 13, 2015 This thesis prepared by Jacob Bobrowski entitled Generalized Catalan Numbers and Some Divisibility Properties is approved in partial fulfillment of the requirements for the degree of Master of Science – Mathematical Sciences Department of Mathematical Sciences Peter Shive, Ph.D. -
The Q , T -Catalan Numbers and the Space of Diagonal Harmonics
The q, t-Catalan Numbers and the Space of Diagonal Harmonics with an Appendix on the Combinatorics of Macdonald Polynomials J. Haglund Department of Mathematics, University of Pennsylvania, Philadel- phia, PA 19104-6395 Current address: Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104-6395 E-mail address: [email protected] 1991 Mathematics Subject Classification. Primary 05E05, 05A30; Secondary 05A05 Key words and phrases. Diagonal harmonics, Catalan numbers, Macdonald polynomials Abstract. This is a book on the combinatorics of the q, t-Catalan numbers and the space of diagonal harmonics. It is an expanded version of the lecture notes for a course on this topic given at the University of Pennsylvania in the spring of 2004. It includes an Appendix on some recent discoveries involving the combinatorics of Macdonald polynomials. Contents Preface vii Chapter 1. Introduction to q-Analogues and Symmetric Functions 1 Permutation Statistics and Gaussian Polynomials 1 The Catalan Numbers and Dyck Paths 6 The q-Vandermonde Convolution 8 Symmetric Functions 10 The RSK Algorithm 17 Representation Theory 22 Chapter 2. Macdonald Polynomials and the Space of Diagonal Harmonics 27 Kadell and Macdonald’s Generalizations of Selberg’s Integral 27 The q,t-Kostka Polynomials 30 The Garsia-Haiman Modules and the n!-Conjecture 33 The Space of Diagonal Harmonics 35 The Nabla Operator 37 Chapter 3. The q,t-Catalan Numbers 41 The Bounce Statistic 41 Plethystic Formulas for the q,t-Catalan 44 The Special Values t = 1 and t =1/q 47 The Symmetry Problem and the dinv Statistic 48 q-Lagrange Inversion 52 Chapter 4. -
Convergence of a Catalan Series Thomas Koshy and Zhenguang Gao
Convergence of a Catalan Series Thomas Koshy and Zhenguang Gao Thomas Koshy ([email protected]) received his Ph.D. in Algebraic Coding Theory from Boston University. His interests include algebra, number theory, and combinatorics. Zhenguang Gao ([email protected]) received his Ph.D. in Applied Mathematics from the University of South Carolina. His interests include information science, signal processing, pattern recognition, and discrete mathematics. He currently teaches computer science at Framingham State University. The well known Catalan numbers Cn are named after Belgian mathematician Eugene Charles Catalan (1814–1894), who found them in his investigation of well-formed sequences of left and right parentheses. As Martin Gardner (1914–2010) wrote in Scientific American [2], they have the propensity to “pop up in numerous and quite unexpected places.” They occur, for example, in the study of triangulations of con- vex polygons, planted trivalent binary trees, and the moves of a rook on a chessboard [1, 2, 3, 4, 6]. D 1 2n The Catalan numbers Cn are often defined by the explicit formula Cn nC1 n , ≥ C j 2n where n 0 [1, 4, 6]. Since .n 1/ n , it follows that every Catalan number is a positive integer. The first five Catalan numbers are 1, 1, 2, 5, and 14. Catalan num- D 4nC2 D bers can also be defined by the recurrence relation CnC1 nC2 Cn, where C0 1. So lim CnC1 D 4. n!1 Cn Here we study the convergence of the series P1 1 and evaluate the sum. Since nD0 Cn n CnC1 P1 x lim D 4, the ratio test implies that the series D converges for jxj < 4. -
~Umbers the BOO K O F Umbers
TH E BOOK OF ~umbers THE BOO K o F umbers John H. Conway • Richard K. Guy c COPERNICUS AN IMPRINT OF SPRINGER-VERLAG © 1996 Springer-Verlag New York, Inc. Softcover reprint of the hardcover 1st edition 1996 All rights reserved. No part of this publication may be reproduced, stored in a re trieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Published in the United States by Copernicus, an imprint of Springer-Verlag New York, Inc. Copernicus Springer-Verlag New York, Inc. 175 Fifth Avenue New York, NY lOOlO Library of Congress Cataloging in Publication Data Conway, John Horton. The book of numbers / John Horton Conway, Richard K. Guy. p. cm. Includes bibliographical references and index. ISBN-13: 978-1-4612-8488-8 e-ISBN-13: 978-1-4612-4072-3 DOl: 10.l007/978-1-4612-4072-3 1. Number theory-Popular works. I. Guy, Richard K. II. Title. QA241.C6897 1995 512'.7-dc20 95-32588 Manufactured in the United States of America. Printed on acid-free paper. 9 8 765 4 Preface he Book ofNumbers seems an obvious choice for our title, since T its undoubted success can be followed by Deuteronomy,Joshua, and so on; indeed the only risk is that there may be a demand for the earlier books in the series. More seriously, our aim is to bring to the inquisitive reader without particular mathematical background an ex planation of the multitudinous ways in which the word "number" is used. -
Problem Solving and Recreational Mathematics
Problem Solving and Recreational Mathematics Paul Yiu Department of Mathematics Florida Atlantic University Summer 2012 Chapters 1–44 August 1 Monday 6/25 7/2 7/9 7/16 7/23 7/30 Wednesday 6/27 *** 7/11 7/18 7/25 8/1 Friday 6/29 7/6 7/13 7/20 7/27 8/3 ii Contents 1 Digit problems 101 1.1 When can you cancel illegitimately and yet get the cor- rectanswer? .......................101 1.2 Repdigits.........................103 1.3 Sortednumberswithsortedsquares . 105 1.4 Sumsofsquaresofdigits . 108 2 Transferrable numbers 111 2.1 Right-transferrablenumbers . 111 2.2 Left-transferrableintegers . 113 3 Arithmetic problems 117 3.1 AnumbergameofLewisCarroll . 117 3.2 Reconstruction of multiplicationsand divisions . 120 3.2.1 Amultiplicationproblem. 120 3.2.2 Adivisionproblem . 121 4 Fibonacci numbers 201 4.1 TheFibonaccisequence . 201 4.2 SomerelationsofFibonaccinumbers . 204 4.3 Fibonaccinumbersandbinomialcoefficients . 205 5 Counting with Fibonacci numbers 207 5.1 Squaresanddominos . 207 5.2 Fatsubsetsof [n] .....................208 5.3 Anarrangementofpennies . 209 6 Fibonacci numbers 3 211 6.1 FactorizationofFibonaccinumbers . 211 iv CONTENTS 6.2 TheLucasnumbers . 214 6.3 Countingcircularpermutations . 215 7 Subtraction games 301 7.1 TheBachetgame ....................301 7.2 TheSprague-Grundysequence . 302 7.3 Subtraction of powers of 2 ................303 7.4 Subtractionofsquarenumbers . 304 7.5 Moredifficultgames. 305 8 The games of Euclid and Wythoff 307 8.1 ThegameofEuclid . 307 8.2 Wythoff’sgame .....................309 8.3 Beatty’sTheorem . 311 9 Extrapolation problems 313 9.1 Whatis f(n + 1) if f(k)=2k for k =0, 1, 2 ...,n? . 313 1 9.2 Whatis f(n + 1) if f(k)= k+1 for k =0, 1, 2 ...,n? .