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RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY (Collected papers) Edited by ZHANG WENPENG Department of Mathematics Northwest University Xi’an, P. R. China Hexis 2004 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY (Collected papers) Edited by ZHANG WENPENG Department of Mathematics Northwest University Xi’an, P. R. China Hexis 2004 This book can be ordered in a paper bound reprint from: Books on Demand ProQuest Information & Learning (University of Microfilm International) 300 N. Zeeb Road P.O. Box 1346, Ann Arbor MI 48106-1346, USA Tel.: 1-800-521-0600 (Customer Service) http://wwwlib.umi.com/bod/search/basic Peer Reviewers: A. W. Vyawahare, H. O. D. Mathematics Department, M. M. College Of Science, Umred Road, Sakkardara, Nagpur University, Nagpur, PIN :- 440009, India. K. M. Purohit, H. O. D. Mathematics Department, V.M.V. Com., J.M.T. Arts & J.J.P. Science College, Wardhaman Nagar, Nagpur University, Nagpur, PIN : – 440008 , India. Dr. (Mrs.) W.B.Vasantha Kandasamy, Department of Mathematics, Indian Institute of Technology, IIT Madras, Chennai - 600 036, India. Copyright 2004 by Hexis (Phoenix, USA) and Zhang Wenpeng, and Authors. Many books one can download from the Digital Library: http://www.gallup.unm.edu/~smarandache/eBooks-otherformats.htm ISBN: 1-931233-88-8 ISBN-13: 978-1-931233-88-0 Standard Address Number: 297-5092 Printed in the United States of America Thisbookisdedicatedtothe memoryofFlorentin Smarandache,wholisted manynewandunsolved problemsinnumbertheory. Contents Dedication v Preface xi (cid:0) Anarithmeticfunctionandtheprimitivenumberofpower 1 ZhangWenpeng (cid:0) (cid:1) Ontheprimitivenumbersofpower and -powerroots 5 YiYuan,LiangFangchi Meanvalueonthepseudo-Smarandachesquarefreefunction 9 LiuHuaning,GaoJing (cid:1) Ontheadditive -thpowercomplements 13 XuZhefeng (cid:2) OntheSmarandachepseudo-multiplesof sequence 17 WangXiaoying Anarithmeticfunctionandthedivisorproductsequences 21 ZhangTianping TheSmarandacheirrationalrootsievesequences 27 ZhangXiaobeng,LouYuanbing Anumbertheoreticfunctionanditsmeanvalue 33 LvChuan (cid:0) Ontheprimitivenumbersofpower anditstriangleinequality 37 DingLiping TheadditiveanalogueofSmarandachesimplefunction 39 ZhuMinhui (cid:1) Onthe -powercomplementsequence 43 YaoWeili Ontheinferiorandsuperiorfactorialpartsequences 47 LiJie Anumbertheoreticfunctionanditsmeanvalue 49 GaoNan viii RESEARCHONSMARANDACHEPROBLEMSINNUMBERTHEORY Onthegeneralizedconstructiveset 53 Gousu Ontheinferiorandsuperiorprimepart 57 LouYuanbing (cid:1) Identitiesonthe -powercomplements 61 ZhangWenpeng Ontheasymptoticpropertyofdivisorfunctionforadditivecomplements 65 YiYuan,LiangFangchi MeanvalueontwoSmarandache-typemultiplicativefunctions 69 LiuHuaning,GaoJing OntheSmarandacheceilfunctionandthenumberofprimefactors 73 XuZhefeng Onthemeanvalueofanarithmeticalfunction 77 WangXiaoying Twoasymptoticformulaeonthedivisorproductsequences 81 ZhangTianping OntheSmarandachepseudo-evennumberSequence 85 ZhangXiaobeng,LouYuanbing Onthemeanvalueofanarithmeticalfunction 89 LvChuan Anarithmeticalfunctionanditscubiccomplements 93 DingLiping Onthesymmetricsequenceanditssomeproperties 97 ZhuMinhui TheadditiveanalogueofSmarandachefunction 99 YaoWeili AnasymptoticformulaonSmarandacheceilfunction 103 LiJie Ahybridnumbertheoreticfunctionanditsmeanvalue 107 GaoNan OntheSmarandachepseudo-number 111 LouYuanbing Severalasymptoticformulaeonanewarithmeticalfunction 115 GuoJinbao and HeYanfeng (cid:1) OntheSmarandachefunctionandthe -throotsofapositiveinteger 119 LiHailong and ZhaoXiaopeng Contents ix OnadualofthePseudoSmarandacheFunctionanditsAsymptoticFormula 123 LiuDuansen and YangCundian (cid:0) Theprimitivenumbersofpower anditsasymptoticproperty 129 LiangFangchi,YiYuan SomeAsymptoticpropertiesinvolvingtheSmarandacheceilfunction 133 HeXiaolin and GuoJinbao AsymptoticformulaeofSmarandache-typemultiplicativeFunctions 139 YangCundian and LiChao (cid:1) Ontheintegerpartofthe -throotofapositiveinteger 143 YangMingshun and LiHailong Ontheadditivecubiccomplements 147 LiangFangchi,YiYuan Anarithmeticalfunctionanditshybridmeanvalue 151 LiChao and LiJunzhuang (cid:1) Onthe -thpowerfreesievesequence 155 GuoJinbao and ZhaoXiqing OnanewSmarandachesequence 159 ZhaoXiaopeng and YangMingshun OnsomeasymptoticformulaeinvolvingSmarandachemultiplicativefunctions 163 LiJunzhuang and LiuDuansen Preface Arithmetic is where numbers run across your mind looking for the answer. Arithmetic is like numbers spinning in your head faster and faster until you blow up with the answer. KABOOM!! Then you sit back down and begin the next problem. (Alexander Nathanson) Number theory is an ancient subject, but we still cannot answer many simplest and most natural questions about the integers. Some old problems have been solved, but more arise. All the research for these ancient or new problems implicated and are still promoting the development of number theory and mathematics. American-Romanian number theorist Florentin Smarandache introduced hundreds of interest sequences and arithmetical functions, and presented many problems and conjectures in his life. In 1991, he published a book named Only problems, Not solutions!. He presented 105 unsolved arithmetical problems and conjectures about these functions and sequences in it. Already many researchers studied these sequences and functions from his book, and obtained important results. This book, Research on Smarandache Problems in Number Theory (Collected papers), contains 41 research papers involving the Smarandache sequences, functions, or problems and conjectures on them. All these papers are original. Some of them treat the mean value or hybrid mean value of Smarandache type functions, like the famous Smarandache function, Smarandache ceil function, or Smarandache primitive function. Others treat the mean value of some famous number theoretic functions acting on the Smarandache sequences, like k-th root sequence, k-th complement sequence, or factorial part sequence, etc. There are papers that study the convergent property of some infinite series involving the Smarandache type sequences. Some of these sequences have been first investigated too. In addition, new sequences as additive complement sequences are first studied in several papers of this book. Most authors of these papers are my students. After this chance, I hope they will be more interested in the mysterious integer and number theory! All the papers are supported by the N. S. F. of P. R. China (10271093). So I would like to thank the Department of Mathematical and Physical Sciences of N. S. F. C. I would also like to thank my students Xu Zhefeng and Zhang Xiaobeng for their careful typeset and design works. My special gratitude is due to all contributors of this book for their great help to the publication of their papers and their detailed comments and corrections. More future papers by my students will focus on the Smarandache notions, such as sequences, functions, constants, numbers, continued fractions, infinite products, series, etc. in number theory! August 10, 2004 Zhang Wenpeng