Research on Smarandache Problems in Number Theory (Collected Papers, 1)

2004·0.93 MB·

by  Zhang Wenpeng

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
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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