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  • Octavian Cira Florentin Smarandache

  • Various Arithmetic Functions and their Applications

    Octavian Cira and Florentin Smarandache

  • 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

  • Various Arithmetic Functions and their

    Applications

    Octavian Cira  Florentin Smarandache

    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

    I

    III

    XI

    XII

    XVII

    Preface

    Contents

    List of Figure

    List of Tables

    Introduction

    1 Prime Numbers 1

    1.1 Generating Primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Sieve of Eratosthenes . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.2 Sieve of Sundaram . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.3 Sieve of Atkin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

    1.2 Primality Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.1 Smarandache Primality Criterion . . . . . . . . . . . . . . . . . . . 4

    1.3 Luhn primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2.5.1 Factorions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.5.2 Double Factorions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.5.3 Triple Factorials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.5.4 Factorial Primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

    2.6 Digital Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.7 Sum–Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.8 Code Puzzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 2.9 Pierced Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 2.10 Divisor Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 2.11 Proper Divisor Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.12 n – Multiple Power Free Sieve . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.13 Irrational Root Sieve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 2.14 Odd Sieve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 2.15 n – ary Power Sieve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 2.16 k – ary Consecutive Sieve . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 2.17 Consecutive Sieve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 2.18 Prime Part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

    2.18.1 Inferior and Superior Prime Part . . . . . . . . . . . . . . . . . . . . 67 2.18.2 Inferior and Superior Fractional Prime Part . . . . . . . . . . . . . 69

    2.19 Square Part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 2.19.1 Inferior and Superior Square Part . . . . . . . . . . . . . . . . . . . 70 2.19.2 Inferior and Superior Fractional Square Part . . . . . . . . . . . . . 71

    2.20 Cubic Part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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

    2.23 Smarandache type Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Smarand