University of New Mexico UNM Digital Repository Mathematics and Statistics Faculty and Staff Publications Academic Department Resources 2016 Various Arithmetic Functions and their Applications Florentin Smarandache University of New Mexico, [email protected] Octavian Cira Follow this and additional works at: https://digitalrepository.unm.edu/math_fsp Part of the Algebra Commons, Applied Mathematics Commons, Logic and Foundations Commons, Number Theory Commons, and the Set Theory Commons Recommended Citation Smarandache, Florentin and Octavian Cira. "Various Arithmetic Functions and their Applications." (2016). https://digitalrepository.unm.edu/math_fsp/256 This Book is brought to you for free and open access by the Academic Department Resources at UNM Digital Repository. It has been accepted for inclusion in Mathematics and Statistics Faculty and Staff Publications by an authorized administrator of UNM Digital Repository. For more information, please contact [email protected], [email protected], [email protected]. Octavian Cira Florentin Smarandache Octavian Cira and Florentin Smarandache Various Arithmetic Functions and their Applications Peer reviewers: Nassim Abbas, Youcef Chibani, Bilal Hadjadji and Zayen Azzouz Omar Communicating and Intelligent System Engineering Laboratory, Faculty of Electronics and Computer Science University of Science and Technology Houari Boumediene 32, El Alia, Bab Ezzouar, 16111, Algiers, Algeria Octavian Cira Florentin Smarandache Various Arithmetic Functions and their Applications PONS asbl Bruxelles, 2016 © 2016 Octavian Cira, Florentin Smarandache & Pons. All rights reserved. This book is protected by copyright. No part of this book may be reproduced in any form or by any means, including photocopying or using any information storage and retrieval system without written permission from the copyright owners Pons asbl Quai du Batelage no. 5 1000 Bruxelles Belgium ISBN 978-1-59973-372-2 Preface Over 300 sequences and many unsolved problems and conjectures related to them are presented herein. These notions, definitions, unsolved problems, questions, theorems corollaries, formulae, conjectures, examples, mathemat- ical criteria, etc. on integer sequences, numbers, quotients, residues, expo- nents, sieves, pseudo-primes squares cubes factorials, almost primes, mobile periodicals, functions, tables, prime square factorial bases, generalized factori- als, generalized palindromes, so on, have been extracted from the Archives of American Mathematics (University of Texas at Austin) and Arizona State Uni- versity (Tempe): "The Florentin Smarandache papers" special collections, Uni- versity of Craiova Library, and Arhivele Statului (Filiala Vâlcea & Filiala Dolj, România). The book is based on various articles in the theory of numbers (starting from 1975), updated many times. Special thanks to C. Dumitrescu and V. Se- leacufrom the University of Craiova (see their edited book "Some Notions and Questions in Number Theory", Erhus Press, Glendale, 1994), M. Bencze, L. Tu- tescu, E. Burton, M. Coman, F. Russo, H. Ibstedt, C. Ashbacher, S. M. Ruiz, J. Sandor, G. Policarp, V. Iovan, N. Ivaschescu, etc. who helped incollecting and editing this material. This book was born from the collaboration of the two authors, which started in 2013. The first common work was the volume "Solving Diophantine Equa- tions", published in 2014. The contribution of the authors can be summarized as follows: Florentin Smarandache came with his extraordinary ability to pro- pose new areas of study in number theory, and Octavian Cira – with his algo- rithmic thinking and knowledge of Mathcad. The work has been edited in LATEX. March 23, 2016 Authors I Contents Preface I Contents III List of Figure XI List of Tables XII Introduction XVII 1 Prime Numbers 1 1.1 Generating Primes . 1 1.1.1SieveofEratosthenes .......................... 1 1.1.2SieveofSundaram............................ 2 1.1.3 Sieve of Atkin . 3 1.2 Primality Criteria . 4 1.2.1 Smarandache Primality Criterion . 4 1.3Luhnprimes.................................... 5 1.3.1 Luhn Primes of First Rank . 6 1.3.2 Luhn Primes of Second Rank . 7 1.4 Endings the Primes . 8 1.5 Numbers of Gap between Primes . 9 1.6 Polynomials Generating Prime Numbers . 10 1.7 Primorial . 14 1.7.1 Double Primorial . 16 1.7.2 Triple Primorial . 18 2 Arithmetical Functions 21 2.1 Function of Counting the Digits . 21 2.2 Digits the Number in Base b ........................... 21 2.3 Prime Counting Function . 21 2.4 Digital Sum . 22 2.4.1 Narcissistic Numbers . 23 2.4.2 Inverse Narcissistic Numbers . 33 III IV CONTENTS 2.4.3 Münchhausen Numbers . 35 2.4.4 Numbers with Digits Sum in Ascending Powers . 37 2.4.5 Numbers with Digits Sum in Descending Powers . 41 2.5 Multifactorial . 44 2.5.1Factorions................................. 45 2.5.2 Double Factorions . 47 2.5.3 Triple Factorials . 49 2.5.4 Factorial Primes . 51 2.6 Digital Product . 55 2.7Sum–Product................................... 57 2.8CodePuzzle.................................... 60 2.9 Pierced Chain . 60 2.10 Divisor Product . 60 2.11 Proper Divisor Products . 61 2.12n–MultiplePowerFreeSieve.......................... 61 2.13 Irrational Root Sieve . 62 2.14OddSieve...................................... 63 2.15 n –aryPowerSieve................................ 63 2.16 k – ary Consecutive Sieve . 65 2.17 Consecutive Sieve . 66 2.18PrimePart..................................... 67 2.18.1InferiorandSuperiorPrimePart.................... 67 2.18.2 Inferior and Superior Fractional Prime Part . 69 2.19SquarePart..................................... 70 2.19.1 Inferior and Superior Square Part . 70 2.19.2 Inferior and Superior Fractional Square Part . 71 2.20CubicPart..................................... 71 2.20.1 Inferior and Superior Cubic Part . 71 2.20.2 Inferior and Superior Fractional Cubic Part . 73 2.21 Factorial Part . 74 2.21.1 Inferior and Superior Factorial Part . 74 2.22 Function Part . 76 2.22.1 Inferior and Superior Function Part . 76 2.22.2 Inferior and Superior Fractional Function Part . 77 2.23SmarandachetypeFunctions.......................... 78 2.23.1 Smarandache Function . 78 2.23.2 Smarandache Function of Order k . 79 2.23.3 Smarandache–Cira Function of Order k . 81 2.24 Smarandache–Kurepa Functions . 82 2.24.1 Smarandache–Kurepa Function of Order 1 . 82 2.24.2 Smarandache–Kurepa Function of order 2 . 83 2.24.3 Smarandache–Kurepa Function of Order 3 . 84 2.25 Smarandache–Wagstaff Functions . 86 CONTENTS V 2.25.1 Smarandache–Wagstaff Function of Order 1 . 86 2.25.2 Smarandache–Wagstaff Function of Order 2 . 87 2.25.3 Smarandache–Wagstaff Function of Order 3 . 87 2.26 Smarandache Near to k–Primorial Functions . 89 2.26.1 Smarandache Near to Primorial Function . 89 2.26.2 Smarandache Near to Double Primorial Function . 89 2.26.3 Smarandache Near to Triple Primorial Function . 90 2.27 Smarandache Ceil Function . 90 2.28 Smarandache–Mersenne Functions . 91 2.28.1 Smarandache–Mersenne Left Function . 91 2.28.2 Smarandache–Mersenne Right Function . 92 2.29 Smarandache–X-nacci Functions . 93 2.29.1 Smarandache–Fibonacci Function . 93 2.29.2 Smarandache–Tribonacci Function . 93 2.29.3 Smarandache–Tetranacci Function . 94 2.30Pseudo–SmarandacheFunctions........................ 94 2.30.1 Pseudo–Smarandache Function of the Order 1 . 94 2.30.2 Pseudo–Smarandache Function of the Order 2 . 97 2.30.3 Pseudo–Smarandache Function of the Order 3 . 101 2.30.4 Alternative Pseudo–Smarandache Function . 104 2.30.5 General Smarandache Functions . 105 2.31 Smarandache Functions of the k-thKind................... 105 2.31.1 Smarandache Function of the First Kind . 105 2.31.2 Smarandache Function of the Second Kind . 105 2.31.3 Smarandache Function of the Third Kind . 105 2.32 The Generalization of the Factorial . 106 2.32.1 Factorial for Real Numbers . 106 2.32.2 Smarandacheial . 108 2.33 Analogues of the Smarandache Function . 112 2.34 Power Function . 113 2.34.1 Power Function of Second Order . 113 2.34.2 Power Function of Third Order . 114 3 Sequences of Numbers 115 3.1 Consecutive Sequence . 115 3.2 Circular Sequence . 120 3.3 Symmetric Sequence . 123 3.4 Deconstructive Sequence . 123 3.5 Concatenated Sequences . 125 3.5.1 Concatenated Prime Sequence . 125 3.5.2 Back Concatenated Prime Sequence . 127 3.5.3 Concatenated Fibonacci Sequence . 129 3.5.4 Back Concatenated Fibonacci Sequence . 129 VI CONTENTS 3.5.5 Concatenated Tetranacci Sequence . 130 3.5.6 Concatenated Mersenne Sequence . 132 3.5.7 Concatenated 6k − 5Sequence..................... 134 3.5.8 Concatenated Square Sequence . 136 3.5.9 Back Concatenated Square Sequence . 137 3.6 Permutation Sequence . 138 3.7 Generalized Permutation Sequence . 139 3.8 Combinatorial Sequences . 139 3.9 Simple Numbers . 141 3.10 Pseudo–Smarandache Numbers . 144 3.10.1 Pseudo–Smarandache Numbers of First Kind . 145 3.10.2 Pseudo–Smarandache Numbers of Second Kind . 149 3.10.3 Pseudo–Smarandache Numbers of Third Kind . 152 3.11 General Residual Sequence . 156 3.12 Goldbach–Smarandache Table . 157 3.13 Vinogradov–Smarandache Table . 160 3.14 Smarandacheian Complements . 165 3.14.1 Square Complements . 165 3.14.2 Cubic Complements . 165 3.14.3 m−power Complements . 166 3.15 m−factorialComplements........................... 166 3.16 Prime Additive Complements . ..