Research on Smarandache Problems in Number Theory, edited by Zhang Wenpeng

8/14/2019 Research on Smarandache Problems in Number Theory, edited by Zhang Wenpeng

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RESEARCH ON

SMARANDACHE PROBLEMS

IN NUMBER THEORY

(Collected papers)

Edited by

ZHANG WENPENG

Department of Mathematics

Northwest University

Xian, P. R. China

Hexis

2004


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RESEARCH ON

SMARANDACHE PROBLEMS

IN NUMBER THEORY(Collected papers)

Edited by

ZHANG WENPENG

Department of Mathematics

Northwest University

Xian, P. R. China

Hexis

2004


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Contents

Dedication v

Preface xi

An arithmetic function and the primitive number of power

1Zhang Wenpeng

On the primitive numbers of power

and

-power roots 5Yi Yuan, Liang Fangchi

Mean value on the pseudo-Smarandache squarefree function 9Liu Huaning, Gao Jing

On the additive

-th power complements 13Xu Zhefeng

On the Smarandache pseudo-multiples of sequence 17Wang Xiaoying

An arithmetic function and the divisor product sequences 21Zhang Tianping

The Smarandache irrational root sieve sequences 27Zhang Xiaobeng, Lou Yuanbing

A number theoretic function and its mean value 33Lv Chuan


and its triangle inequality 37Ding Liping

The additive analogue of Smarandache simple function 39Zhu Minhui

On the

-power complement sequence 43Yao Weili

On the inferior and superior factorial part sequences 47Li Jie

A number theoretic function and its mean value 49Gao Nan


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viii RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY

On the generalized constructive set 53Gou su

On the inferior and superior prime part 57Lou Yuanbing

Identities on the

-power complements 61Zhang Wenpeng

On the asymptotic property of divisor function for additive complements 65Yi Yuan, Liang Fangchi

Mean value on two Smarandache-type multiplicative functions 69

Liu Huaning, Gao Jing

On the Smarandache ceil function and the number of prime factors 73Xu Zhefeng

On the mean value of an arithmetical function 77Wang Xiaoying

Two asymptotic formulae on the divisor product sequences 81Zhang Tianping

On the Smarandache pseudo-even number Sequence 85Zhang Xiaobeng, Lou Yuanbing

On the mean value of an arithmetical function 89Lv Chuan

An arithmetical function and its cubic complements 93Ding Liping

On the symmetric sequence and its some properties 97Zhu Minhui

The additive analogue of Smarandache function 99Yao Weili

An asymptotic formula on Smarandache ceil function 103Li Jie

A hybrid number theoretic function and its mean value 107Gao Nan

On the Smarandache pseudo-number 111Lou Yuanbing

Several asymptotic formulae on a new arithmetical function 115Guo Jinbao and He Yanfeng

On the Smarandache function and the

-th roots of a positive integer 119 Li Hailong and Zhao Xiaopeng


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Contents ix

On a dual of the Pseudo Smarandache Function and its Asymptotic Formula 123Liu Duansen and Yang Cundian

The primitive numbers of power

and its asymptotic property 129Liang Fangchi, Yi Yuan

Some Asymptotic properties involving the Smarandache ceil function 133 He Xiaolin and Guo Jinbao

Asymptotic formulae of Smarandache-type multiplicative Functions 139Yang Cundian and Li Chao

On the integer part of the

-th root of a positive integer 143

Yang Mingshun and Li Hailong

On the additive cubic complements 147Liang Fangchi, Yi Yuan

An arithmetical function and its hybrid mean value 151 Li Chao and Li Junzhuang

On the -th power free sieve sequence 155Guo Jinbao and Zhao Xiqing

On a new Smarandache sequence 159Zhao Xiaopeng and Yang Mingshun

On some asymptotic formulae involving Smarandache multiplicative functions 163

Li Junzhuang and Liu Duansen


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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 naturalquestions 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 numbertheory and mathematics.

American-Romanian number theorist Florentin Smarandache introduced hundreds of interestsequences 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 unsolvedarithmetical 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 41research 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, orSmarandache 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 inseveral papers of this book.

Most authors of these papers are my students. After this chance, I hope they will be more interestedin 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.


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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 tothe 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


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AN ARITHMETIC FUNCTION AND THE PRIMITIVE

NUMBER OF POWER

Zhang WenpengResearch Center for Basic Science, Xian Jiaotong University, Xian, Shaanxi, P.R.China

[email protected]

Abstract For any fixed prime

, we define

"

$ &

( ) 2

$ 4

6 8 (

) "

E &

) 2

E I Q 4

T

The main purpose of this paper is to study the mean value properties of)

,

and give an interesting asymptotic formula for it.

Keywords: Primitive number; Mean value; Asymptotic formula.

W

1. Introduction

Let be a prime,X

be any positive integer, we define two arithmetic func-

tions as following:

Y ` b

X c e g i p q

r t

g u v X x

r

b

g c u i

v

` b

X c e g i p q

t

X v X x

X

In problem 49 and 68 of reference [1], Professor F.Smarandache asked us to

study the properties of these two arithmetic functions. About these problems,

many scholars showed great interests in them (See references [2], [3]). But it

seems that no one knows the relationship between these two arithmetic func-tions before. In this paper, we shall use the elementary methods to study the

mean value properties ofv

` b Y ` b

X c c, and give an interesting asymptotic for-

mula for it. That is , we shall prove the following conclusion:

Theorem. For any fixed prime and any real number , we have the

asymptotic formula

r

v

` b Y ` b

X c c e

b

c j

l n

Taking e

,

in the theorem, we may immediately obtain the following


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2 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY

Corollary. For any real number

, we have the asymptotic formula

r

v

j

b Y

j

b

X c c e

m l n

i

r

v

b Y

b

X c c e

m l n

W

2. One simple lemma

To complete the proof of the theorem, we need the following simple lemma:

Lemma. For any fixed prime and real number

, we have

|

}

j

|

e

j

b

c

l ~

j

Proof. First we come to calculate

e

|

r

}

j

|

Note that the identities

e

r

|

}

j

|

r

|

}

j

|

e

X

j

r

r

|

b

}

c

j

}

j

|

e

X

j

r

r

|

}

|

i

and

j

e

~

X

j

r

r

|

}

|

e

j

X

j

X

j

r

j

r

|

}

|

r

|

j

}

|

e

j

X

j

X

j

r

j

r

|

j

|

X

j

X

j

b

X c

r

j

e

b

r

c

r

r

X

j

b

X

j

X c

r

j

So we have

e

~

b

r

c

r

r

X

j

b

X

j

X c

r

j

n

`

j


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An arithmetic function and the primitive number of power 3

e

b

c j

b

r

c

r

j

b

c

X

j

b

X

j

X c

r

b

c j

Then we can immediately obtain

|

}

j

|

e

b

c

j

b

c

l~

j

e

j

b

c

l ~

j

This completes the proof of the Lemma.W

3. Proof of the Theorem

In this section, we shall use the above Lemma to complete the proof of the

Theorem. From the definition ofY

`b

X c

andv

`b

X c

, we may immediately get

r

v

` b Y ` b

X c c

e

`

}

j

e

`

`

}

j

e

`

}

j

`

`

e

`

}

j

`

`

b

x c

e

`

}

j

`

b

x c

`

e

`

}

j

`

`

I Q

e

`

}

j

|

|

l

b

c

e

|

`

}

j

|

|

`

}

j

|

l

|

`

}

j

e

|

`

}

j

|

l n

e

~

j

b

c

l ~

j

l n

e

b

c j

l n

This completes the proof of the Theorem.


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References

[1] Smarandache F. Only problems, not Solutions. Chicago: Xiquan Publ.

House, 1993.

[2] Liu Hongyan and Zhang Wenpeng. A number theoretic function and its

mean value property. Smarandache Notions Journal, 2002, 13: 155-159.

[3] Zhang Wenpeng and Liu Duansen. On the primitive number of power

and its asymptotic property. Smarandache Notions Journal, 2002, 13: 173-175.

[4] Apostol T M. Introduction to Analytic Number Theory. New York:

Springer-Verlag, 1976.


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ON THE PRIMITIVE NUMBERS OF POWER

AND

-POWER ROOTS

Yi YuanResearch Center for Basic Science, Xian Jiaotong University, Xian, Shaanxi, P.R.China

[email protected]

Liang FangchiSchool of Science, Air Force Engineering University, Xian, Shaanxi, P.R.China

[email protected]

Abstract Let be a prime,

be any positive integer,

denotes the smallest integer

I

, where

$

&

(

. In this paper, we study the mean value properties

of

)

$

, where)

$

is the superior integer part of -power roots, and give an

interesting asymptotic formula for it.

Keywords: Primitive numbers of power ; -power roots; Asymptotic formula.

W

1. Introduction and results

Let be a prime,X

be any positive integer,Y ` b

X cdenotes the smallest

integer such thatY ` b

X c uis divisible by r . For example,

Y

b

c e ,

Y

b

c e

,Y

b

c e ,Y

b -

c e , . In problem 49 of book [1], Professor F.

Smarandache ask us to study the properties of the sequence

Y ` b

X c . About

this problem, Professor Zhang and Liu in [2] have studied it and obtained an

interesting asymptotic formula. That is, for any fixed prime and any positive

integerX

,

Y ` b

X c e

b

c X

l

X

For any fixed positive intger

, letv

r

denotes the superior integer part of

-

power roots, that is,v

e ,

,

v

j

e ,

v

j

e ,

. In problem 80 of book

[1], Professor F. Smarandache ask us to study the properties of the sequence

v

r

. About this problem, the author of [3] have studied it and obtained an

interesting asymptotic formula. That is, for any real number

,

r

b

v

r

c e

b

c

l

i


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where

b

X cdenotes the numbers of all prime divisor of

X,

be a computable

constant.

In this paper, we will use the elementary method to study the asymptotic

properties ofY ` b

v

r

cin the following form:

r

t

Y`

b

v

r

c

Y`

b

v

r

c

t

i

where

be a positive real number, and give an interesting asymptotic formula

for it. In fact, we shall prove the following result:

Theorem. For any real number , let be a prime and X be any

positive integer. Then we have the asymptotic formula

r

t

Y ` b

v

r

c

Y ` b

v

r

c

t

e

Q

l

i

where l denotes the l -constant depending only on parameter

.

W


In this section, we shall complete the proof of the theorem. First we need

following one simple Lemma. That is,

Lemma. Let be a prime andX

be any positive integer, then we have

t

Y ` b

X

c

Y ` b

X c

t

e

i if r t t g u

iotherwise

i

whereY ` b

X c e g , r g u denotes that r t g u and r g u .

Proof. Now we will discuss it in two cases.b

p c LetY ` b

X c e g , if r g u , then we have r t g u and r g u . From the

definition ofY

`b

X cwe have r

b

g

c u, r

b

g

c u,

, r

b

g

c uand r t

b

g

c u, so

Y ` b

X

c e g

, then we get

t

Y`

b

X

c

Y`

b

X c

t

e

(1)

b

p p c

Let

Y ` b

X c e g

, if

r

t

g u

and

r

t

g u

, then we have

Y ` b

X

c e g

,so

t

Y ` b

X

c

Y ` b

X c

t

e

(2)

Combining (1) and (2), we can easily get

t

Y ` b

X

c

Y ` b

X c

t

e

iif r

g u

i otherwise

This completes the proof of Lemma.

Now we use above Lemma to complete the proof of Theorem. For any real

number

, let

be a fixed positive integer such that

b

c

,


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and -power roots 7

then from the definition ofY ` b

X cand the Lemma we have

r

t

Y ` b

v

r

c

Y ` b

v

r

c

t (3)

e

r

t

Y ` b

v

r

c

Y ` b

v

r

c

t

r

t

Y ` b

v

r

c

Y ` b

v

r

c

t

e

t

Y ` b

c

Y ` b

c

t

r

t

Y ` b

v

r

c

Y ` b

v

r

c

t

e

t

Y ` b

c

Y ` b

c

t

e

Q

`

l

b

c (4)

whereY ` b

c e g. Note that if

g u, then we have (see reference [4],

Theorem 1.7.2)

e

g

e

g

e g

l n

`

g

e

g

l

g

(5)

From (4), we can deduce that

g e

b

c

l

(6)

So that

g

b

c

Q

l

i if

Q

Note that for any fixed positive integer

, if there has oneg

such that

g u,

then

b

g

c u,

b

g

c u,

,

b

g

c u. Hence there have

times ofg

such that

e

`

in the interval

g

b

c

Q

l


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Then from this and (3), we have

r

t

Y ` b

v

r

c

Y ` b

v

r

c

t

e

Q

`

l

b

c

e

b

c

Q

l

l

b

c

e

Q

l

This completes the proof of Theorem.

References


House, 1993.

[2] Zhang Wenpeng and Liu Duansen. primitive numbers of power and its

asymptotic property, Smaramche Notions Journal 2002, 13: 173-175.

[3] Yang Hai. Yanan University masters degree dissertion, 2004, 19-23.

[4] Pan Chengdong and Pan Chengbiao, The Elementary number Theory,

Beijing University Press Beijing, 2003.[5] Apostol T M. Introduction to Analytic Number Theory. New York:



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MEAN VALUE ON THE PSEUDO-SMARANDACHE

SQUAREFREE FUNCTION

Liu HuaningDepartment of Mathematics, Northwest University, Xian, Shaanxi, P.R.China

[email protected]

Gao JingSchool of Science, Xian Jiaotong University, Xian, Shaanxi, P.R.China

[email protected]

Abstract For any positive integer

, the pseudo-Smarandache squarefree function

is defined as the least positive integer

such that

$

is divisible by

. In this

paper, we study the mean value of

, and give a few asymptotic formulae.

Keywords: Pseudo-Smarandache squarefree function; Mean value; Asymptotic formula.

W

1. Introduction

According to [1], the pseudo-Smarandache squarefree function b

X c is

defined as the least positive integerg

such thatg

r is divisible byX

. It is

obvious that

b

c e . For

X , Maohua Le [1] obtained that

b

X c e

j

i(1)

where

, j

,

, are distinct prime divisors ofX

. Also he showed that

r

b

b

X c c

i v i v

is divergence.

In this paper, we study the mean value of

b

X c, and give a few asymptotic

formulae. That is, we shall prove the following:

Theorem 1. For any real numbers }i

with

}

and }

, we have

r

|

b

X c

X

e

b

c

b

}

c

b

}

c

`

|

i


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where

b

c is the Riemann zeta function,

`

denotes the product over all prime

numbers.

Theorem 2. For any real numbers }

and

, we have

r

|

b

X c e

b

}

c

|

b

c

b

}

c

`

|

b

c

ln

|

Q

Noting that

r

b

X c e

l

b

cand

|

I

}

b

}

c e , so from

Theorem 2 we immediately have the limit

|

I

}

`

|

b

c

e

b

c

W

2. Proof of the theorems

Now we prove the theorems. For any real numbers } ,

with

}

and

}

, let

q

b

c e

r

|

b

X c

X

From (1) and the Euler product formula [2] we have

q

b

c e

`

|

|

j

e

`

`

`

e

`

~

`

`

|

e

b

c

b

}

c

b

}

c

`

|

This proves Theorem 1.

For any real numbers }

and

, it is obvious that

t

|

b

X c

t

X

|

and

r

|

b

X c

X

}

i

where is the real part of

. So by Perron formula [3] we can get

r

|b

X c

X

e

p

q

b

c

d

l ~

b

c

l

b

c

i

l

b

c

i

t t

t t

i


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Mean value on the pseudo-Smarandache squarefree function 11

where

is the nearest integer to

, and t t

t t

e

t

t . Taking

e

,

e

}

j

and

in the above, then we have

r

|

b

X c e

p

|

|

q

b

c

d

l~

|

Now we move the integral line from } j

p

to } j

p

. This time,

the function

q

b

c

have a simple pole point at e } with residue

b

}

c

|

b

c

b

}

c

`

|

b

c

Now taking

e , then we have

r

|

b

X c e

b

}

c

|

b

c

b

}

c

`

|

b

c

p

|

Q

|

Q

q

b

c

d

l n

|

Q

e

b

}

c

|

b

c

b

}

c

`

|

b

c

l~

q

}

p

|

Q

b

t

t

c

d

ln

|

Q

e

b

}

c

|

b

c

b

}

c

`

|

b

c

l n

|

Q

This completes the proof of Theorem 2.

References

[1] Maohua Le. On the pseudo-Smarandache squarefree function. Smaran-dache Notions Journal, 2002, 13: 229-236.



[3] Chengdong Pan and Chengbiao Pan. Foundation of Analytic Number

Theory. Beijing: Science Press, 1997.


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23/178

ON THE ADDITIVE

-TH POWER COMPLEMENTS

Xu ZhefengDepartment of Mthematics, Northwest University, Xian, Shaanxi, P.R.China

[email protected]

Abstract In this paper, similar to the Smarandache

-th power complements, we defined

the additive -th power complements. Using the elementary method, we study

the mean value properties of the additive square complements, and give some

interesting asymptotic formulae.

Keywords: Additive

-th power complements; Mean value; Asymptotic formula.

W

1. Introduction

For any positive integerX

, the Smarandache

-th power complements

b

X c

is the smallest positive integer such thatX

b

X cis a complete

-th power, see

problem 29 of [1]. Similar to the Smarandache

-th power complements, wedefine the additive

-th power complementsv

b

X cas follows:

v

b

X cis the

smallest nonnegative integer such thatv

b

X c

Xis a complete

-th power.

For example, if e

, we have the additive square complements sequence

v

j

b

X c

b

X e i i cas follows:

i i i

i

-

i i i i

i

i i

-

i i i i

i i .

In this paper, we stdudy the mean value properties ofv

b

X cand

x

b

v

b

X c c,

where xb

X c is the Dirichlet divisor function, and give several interesting asymp-

totic formulae. That is, we shall prove the following conclusion:

Theorem 1. For any real number


r

v

b

X c e

j

-

j

Q

ln

j

Theorem 2. For any real number


r

x

b

v

b

X c c e

l n

Q

i

where is the Euler constant.

W

2. Some lemmas

Before the proof of the theorems, some lemmas will be usefull.


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Lemma 1. For any real number

, we have the asymptotic formula:

r

x

b

c e

b

c

l n

Q

i

where is the Euler constant.

Proof. See reference [2].

Lemma 2. For any real number and any nonnegative arithmetical

function qb

X c with qb

c e

, we have the asymptotic formula:

r

q

b

v

b

X c c e

Q

r

q

b

X c

l

r

n

Q

q

b

X c

!

!

!

i

where # $

denotes the greatest integer less than or equal to

and %b

c e

p

.

Proof. For any real number

, let

be a fixed positive integer such

that

b

c

Noting that ifX

pass through the integers in the interval

i

b

c

, then

v

b

X c pass through the integers in the inteval

i

b

c

and qb

c e

, we can write

r

q

b

v

b

X c c e

r &

q

b

v

b

X c c

r

q

b

v

b

X c c

e

r

q

b

X c

r &

q

b

X c i

where %b

c e

p

. Since e

Q

, so we have

r

q

b

v

b

X c c e

Q

r

q

b

X c

l

r

n

Q

q

b

X c

!

!

!

This proves Lemma 2.

Note: This Lemma is very usefull. Because if we have the mean value for-

mula ofq

b

X c, then from this lemma, we can easily get the mean value formula

of

r

q

b

v

b

X c c.


25/178

On the additive -th power complements 15

W


In this section, we will complete the proof of the theorems. First we prove

Theorem 1. From Lemma 1 and the Euler summation formula (See [3]), let

q

b

X c e X, we have

r

v

b

X c e

Q

r

X

l

r

n

Q

X

!

!

!

e

Q

j

j

j

l n

j

e

j

-

j

Q

l n

j

This proves Theorem 1.

Now we prove Theorem 2. From Lemma 1 and Lemma 2, we have

r

x

b

v

b

X c c

e

Q

r

x

b

X c

l

r

n

Q

x

b

X c

!

!

!

e

Q

l

b

c

l n

Q

e

Q

n

b

c

b

c

l

b

j

c

l n

Q

e

b

c

Q

b

c

Q

l n

Q

Then from the Euler summation formula, we can easily get

r

x

b

v

b

X c c e

l n

Q


26/178


This completes the proof of the theorems.

References

[1] F.Smaradache. Only problems, not solutions, Xiquan Publishing House,

Chicago, 1993.

[2] G.L.Dirichlet. Sur lusage des series infinies dans la theorie des nom-

bres. Crelles Journal, No.18, 1938.

[3] Tom M. Apostol. Introduction to Analytic Number Theory, Springer-

Verlag, New York, 1976, pp54.


27/178

ON THE SMARANDACHE PSEUDO-MULTIPLES

OF ( SEQUENCE

Wang Xiaoying1. Research Center for Basic Science, Xian Jiaotong University, Xian, Shaanxi, P.R.China

2. Department of Mathematics, Northwest University, Xian, Shaanxi, [email protected]

Abstract The main purpose of this paper is to study the mean value properties of the

Smarandache pseudo-multiples of ) number sequence, and give an interesting

asymptotic formula for it.

Keywords: Pseudo-multiples of ) numbers; Mean value; Asymptotic formula.

W

1. Introduction

A number is a pseudo-multiple of

if some permutation of its digits is a

multiple of , including the identity permutation. For example:

i i

i i

i i

i i

-

i

i i i are pseudo-multiple of

numbers. Let

de-

notes the set of all the pseudo-multiple of

numbers. In reference [1], Profes-

sor F. Smarandache asked us to study the properties of the pseudo-multiple of

sequence. About this problems, it seems that none had studied it, at least we

have not seen such a paper before. In this paper, we use the elementary method

to study the mean value properties of this sequence, and obtain an interesting

asymptotic formula for it. That is, we shall prove the following:

Theorem. For any real number , we have the asymptotic formula

r 0 1

r

q

b

X c e

r

q

b

X c

l n

3 5 6

3 5

Q

i

where e 7 9

r

t

q

b

X c

t

. Taking qb

Xd c e x

b

X c,

b

X cas the Dirichlet

divisor function and the function of the number of prime factors respectively,

then we have the following:

Corollary 1. For any real number


r 0 1

r

x

b

X c e

b

c

l n

3 5 6

3 5

Q

i

where is the Euler constant,

is any fixed positive number.


28/178


Corollary 2. For any real number


r 0 1

r

b

X c e

l

i

where

is a computable constant.

W


Now we completes the proof of the Theorem. First let

b

c , then

. According to the definition of set

, we

know that the largest number of digits (

) not attribute set

isC

. Infact, in these numbers, there are C one digit, they are

i i i

-

i

i i C i ; There

are C j two digits; The number of the

digits are C

. So the largest number of

digits (

) not attribute set

is C C j

C

e DF

b

C

c

C

. Since

C

C

e

n

C

6

Q

3 G H

6

Q

e

b

c

Q

3 G H

6

Q

e

3 5 6

3 5

Q

So we have,

C

e

l n

3 5 6

3 5

Q

Next, let

denotes the upper bounds of tq

b

X c

t

b

X

c, then

r 0

1

r

q

b

X c e

l n

3 5 6

3 5

Q

Finally, we have

r 0 1

r

q

b

X c e

r

q

b

X c

r 0

1

r

q

b

X c

e

r

q

b

X c

ln

3 5 6

3 5

Q

This proves the Theorem.

Now the Corollary 1 follows from the Theorem, the asymptotic formula

r

x

b

X c e

b

c

l n

Q

(see [2]), and the estimate xb

X c P

(for all

X

). And then, the

Corollary 2 follows from the Theorem, the asymptotic formula

r

b

X c e

l

(See [3]), and the estimate

b

X c P

(for all

X

).


29/178

On the Smarandache pseudo-multiples of sequence 19

References


House, 1993.

[2] Tom M. Apostol. Introduction to Analytic Number Theory. New York:


[3] G.H. Hardy and S. Ramanujan. The normal number of prime factors of

a numberX

. Quart. J. Math. 48(1917), 76-92.


30/178


31/178

AN ARITHMETIC FUNCTION AND THE DIVISOR

PRODUCT SEQUENCES

Zhang TianpingDepartment of Mathematics, Northwest University, Xian, Shaanxi, P.R.China

[email protected]

Abstract Let

be any positive integer, R S

denotes the product of all positive divisors

of

. Let

be a prime,)

denotes the largest exponent (of power

) such

that divisible by

. In this paper, we study the asymptotic properties of the mean

value of)

RS

, and give an interesting asymptotic formula for it.

Keywords: Divisor product sequences; Mean value; Asymptotic formula.

W

1. Introduction

LetX

be any positive integer, U b

X cdenotes the product of all positive

divisors of X . That is, U b

X c e

`

x . For example, U b

c e i U

b

c e

i U

b

c e i U

b -

c e C i Let be a prime,

v

` b

X cdenotes the largest ex-

ponent (of power ) such that

r

t

X. In problem 25 and 68 of reference [1],

Professor F.Smarandache asked us to study the properties of these two arith-

metic functions. About these problems, many scholars showed great interests

in them (see references [2],[3]). But it seems that no one knows the relation-

ship between these two arithmetic functions before. In this paper, we shall

use the elementary methods to study the mean value properties ofv

`b Y

`b

X c c,

and give an interesting asymptotic formula for it. That is , we shall prove the

following conclusion:

Theorem. Let

be a prime, then for any real number

, we have theasymptotic formula

r

v

` b

U

b

X c c e

b

c

b

c

b

c X

`

-

j

b

b

c

`

l

b

Q

c i

where is the Euler constant, and

denotes any fixed positive number.

Taking e

,

in the theorem, we may immediately obtain the following


32/178


Corollary. For any real number


r

v

j

b

U

b

X c c e

l

b

Q

c

r

v

b

U

b

X c c e

C

-

-

l

b

Q

c

W

2. Some lemmas

To complete the proof of the theorem, we need the following simple lem-

mas:

Lemma 1. For any positive integerX

, we have the identity

U

b

X c e X

Se

$ f

i

wherex

b

X cis the divisor function.

Proof. This formula can be immediately got from Lemma 1 of [2].

Lemma 2. For any real number , we have the asymptotic formula

r

r

x

b

X c e

`

`

j

l

b

Q

c i

where

`

denotes the product over all primes, is the Euler constant, and

denotes any fixed positive number.

Proof. Let

e

j

i

b

c e

`

j

Then by the Perron formula

(See Theorem 2 of reference [4]), we may obtain

r

r

x

b

X c e

p

j

b

c

b

c

x

l

b

Q

c i

where

b

c is the Riemann-zeta function.

Moving the integral line from j

p

to j

p

. This time, the function

q

b

c e

j

b

c

b

c

has a second order pole point at e with residue

`

`

j


33/178

An arithmetic function and the divisor product sequences 23

So we have

p

~

Q

Q

Q

Q

j

b

c

b

c

x

e

`

`

j

Note that

p

~

Q

Q

Q

Q

j

b

c

b

c

x

P

Q

From the above we can immediately get the asymptotic formula:

r

r

x

b

X c e

`

`

j

l

b

Q

c

This completes the proof of Lemma 2.

Lemma 3. Let be a prime, then for any real number , we have the

following asymptotic formulae

|

}

|

e

b

c j

l

i(1)

|

}

j

|

e

j

b

c

l~

j

i(2)

|

}

|

e

-

j

b

c

`

l~

(3)

Proof. We only prove formulab

cand

b

c. First we come to calculate

q e

|

r

}

j

|


q

e

r

|

}

j

|

r

|

}

j

|

e

X

j

r

r

|

b

}

c

j

}

j

|

e

X

j

r

r

|

}

|


34/178


and

q

j

e

~

X

j

r

r

|

}

|

f

e

j

X

j

X

j

r

j

r

|

}

|

r

|

j

}

|

e

j

X

j

X

j

r

j

r

|

j

|

X

j

X

j

b

X c

r

j

e

b

r

c

r

r

X

j

b

X

j

X c

r

j

So we have

q e

~

b

r

c

r

r

X

j

b

X

j

X c

r

j

n

`

j

e

b

c j

b

r

c

r

j

b

c

X

j

b

X

j

X c

r

b

c j

Then we can immediately obtain

|

}

j

|

e

b

c j

b

c

l ~

j

e

j

b

c

l~

j

This proves formulab

c.

Now we come to prove formulab

c. Let

%e

|

r

}

|


%

e

r

|

}

|

r

|

}

|

e

X

r

r

|

b

}

c

}

|


35/178

An arithmetic function and the divisor product sequences 25

e

X

r

r

|

j

}

j

}

|

e

X

r

r

r

r

~

j

X

r

r

j

r

r

`

~

-

j

X

j

r

~

X

r

b

r

c

r

r

`

`

e

j

-

b

c j

X X

j

r

b

X c

r

b

c

X

r

b

r

c

b

r

j

c

b

c

r

b

c

Then we have

|

}

|

e

~

j

-

b

cj

b

c

b

c

l ~

e

-

j

b

c

`

l~

This completes the proof of Lemma 3.

W


In this section, we shall use the above lemmas to complete the proof of the

Theorem. From the definition of U b

X cand

v

`b

X c, we may immediately get

r

v

`b

U

b

X c c

e

` g

`

g

b

}

c

}

x

b i

c e

`

b

}

c

}

g

`

`

g

x

b i

c

e

|

`

b

}

c

}

~

|

|

j

ln

Q

e

j

|

`

b

}

c

}

|

|

`

b

}

c

}

j

|

l n

Q

e

j


36/178


p

~

j

b

c

b

c j

l~

j

~

-

j

b

c

`

j

b

c

l ~

ln

Q

e

b

c

b

c

b

c X

`

-

j

b

b

c

`

l n

Q

This completes the proof of the Theorem.

References


House, 1993.

[2] Liu Hongyan and Zhang Wenpeng. On the divisor products and proper

divisor products sequences. Smarandache Notions Journal, 2002, 13: 128-133.



[4] Pan Chengdong and Pan Chengbiao. Elements of the analytic number

Theory. Beijing: Science Press, 1991.


37/178

THE SMARANDACHE IRRATIONAL ROOT SIEVE

SEQUENCES

Zhang XiaobengDepartment of Mathematics, Northwest University, Xian, Shaanxi, P.R.China

[email protected]

Lou YuanbingCollege of Science, Tibet University, Lhasa, Tibet, P.R.China

[email protected]

Abstract In this paper, we use the analytic method to study the mean value properties of

the irrational root sieve sequence, and give an interesting asymptotic formula for

it.

Keywords: Smarandache irrational root sieve; Mean value; Asymptotic formula.

W

1. Introduction

According to reference [1], the definition of Smarandache irrational root

sieve is: from the set of natural numbers (except

and

):

-take off all powers of

,

;


,

;


, ;


,

;


,

;


, ;

and so on (take off all

-powers,

). For example:

i i i

i i

i

i i i

-

i i i C i are all irrational root sieve sequence. Let

denotes the set of all the irrational root sieve. In reference [1], Professor F.

Smarandache asked us to study the properties of the irrational root sieve se-

quence. About this problem, it seems that none had studied it, at least we have

not seen such a paper before. In this paper, we study the mean value of the

irrational root sieve sequence, and give an interesting asymptotic formula for

it. That is, we shall prove the following:


38/178


Theorem. Let xb

X c denote the divisor function. Then for any real number


r 0 1

r

x

b

X c

e

-

j

q

Q

j

j

Q

Q

`

q

b

c

r

q

t

Q

ln

Q

u

v

u

i

where

denotes any fixed positive number, is the Euler constant,

i

j

i

i

`

i

r

i

t

are the computable constants.

W

2. Some Lemmas

To complete the proof of the theorem, we need the following lemmas:

Lemma 1. For any real number , we have the asymptotic formula:

r

x

b

X c e

b

c

l n

Q

u

v

u

i

where

denotes any fixed positive number and is the Euler constant.

Proof. This result may be immediately got from [2].

Lemma 2. For any real number , we have two asymptotic formulae

r x

x

b

X

j

c e

q

j

-

j

q

j

q

l n

Q

v

r

Q

x

b

X

c e

Q

`

Q

j

j

Q

Q

l n

Q

i

where

i

j

i

i

i

j

i

are computable constants.

Proof. Let

q

b

c e

r

x

b

X

j

c

X

i

Reb

c . Then from the Euler product formula [3] and the multiplicative

property ofx

b

X cwe have

q

b

c e

`

j

e

`

j

e

`

~

j


39/178

The Smarandache irrational root sieve sequences 29

e

`

j

e

b

c

b

c

where

b

c is the Riemann zeta-function. By Perron formula [2] with

e

,

e

Q

and

e

j

, we have

r

x

b

X

j

c e

p

b

c

b

c

x

ln

Q

To estimate the main term

p

b

c

b

c

x i

we move the integral line from e

j

p

to e

j

p

. This time, the

function

q

b

c e

b

c

b

c

has a three order pole point at e

with residue

u

~

b

c

b

c

b

c

j

e

j

j

j

i

where

i

j

are the computable constants.

Note that

p

~

Q

Q

Q

Q

b

c

b

c

x P

Q

From above we may immediately get the asymptotic formula:

r

x

b

X

j

c e

j

j

j

ln

Q

That is,

r x

x

b

X

j

c e

q

j

-

j

q

j

q

l n

Q

v

This proves the first formula of Lemma 2.

Similarly, we can deduce the second asymptotic formula of Lemma 2. In

fact let

%

b

c e

r

x

b

X

c

X

i


40/178


Reb

c . Then from the Euler product formula [3] and the multiplicative

property ofx

b

X cwe have

%

b

c e

`

-

j

e

`

j

e

`

~

j

e

`

j

e

`

b

c

j

b

c

`

b

c j

where

b

cis the Riemann zeta-function. Then by Perron formula [2] and the

method of proving the first asymptotic formula of Lemma 2 we may immedi-

ately get

r

x

b

X

c e

`

j

j

l n

Q

That is,

r

Q

x

b

X

c e

Q

`

Q

j

j

Q

Q

l n

Q

i

This proves the Lemma 2.

W


Now we completes the proof of the Theorem. According to the definition of

the set

and the result of Lemma 1 and Lemma 2 , we have

r 0 1

r

x

b

X c

e

r

x

b

X c

r x

x

b

X

j

c

r

Q

x

b

X

c

l

`

3 5

3 5

r

Q

x

b

X

c

!

e

r

x

b

X c

r x

x

b

X

j

c

r

Q

x

b

X

c

l

`

3 5

3 5

Q

!


41/178

The Smarandache irrational root sieve sequences 31

e

~

q

-

j

Q

j

j

Q

Q

`

q

b

c

r

q

t

Q

ln

Q

u

v

u

i

where

(p e i i i

) are computable constants.

References


House, 1993.

[2] Pan Chengdong and Pan Chengbiao. Elements of the analytic numberTheory. Beijing: Science Press, 1991.

[3] Tom M. Apostol. Introduction to Analytic Number Theory. New York:



42/178


43/178

A NUMBER THEORETIC FUNCTION AND ITS MEAN

VALUE

Lv ChuanDepartment of Mathematics, Northwest University, Xian, Shaanxi, P.R.China

[email protected]

Abstract Let

be a prime,

denote the largest exponent of power

which divides

. In this paper, we study the properties of this sequence

, and give an

interesting asymptotic formula for

$

E

.

Keywords: Asymptotic formula; Largest exponent; Mean value.

W

1. Introduction

Let be a prime, ` b

X c denote the largest exponent of power which di-

videsX

. In problem 68 of [1], Professor F.Smarandach asked us to study the

properties of the sequence `

b

X c. This problem is closely related to the factor-

ization of X u . In this paper, we use elementary methods to study the asymptotic

properties of the mean value

r

`

b

X c, and give an interesting asymptotic

formula for it. That is, we will prove the following:

Theorem. Let be a prime, g

be an integer. Then for any real number


r

`

b

X c e

v

` b

g c

l

b

c i

where v` b

g c is a computable constant.

Taking g e i i in the theorem, we may immediately obtain the follow-ing

Corollary. For any real number , we have the asymptotic formula

r

` b

X c e

l

b

j

c

r

j

`

b

X c e

b

c j

l

b

c

r

`

b

X c e

j

-

b

c

l

b

`

c


44/178


W


In this section, we complete the proof of the theorem. In fact, from the

definition of ` b

X cwe have

r

`

b

X c e

`

`

`

}

e

|

3 G H

3 G H

}

`

e

|

3 G H

3 G H

}

|

l

b

c

e

|

3 G H

3 G H

}

|

l

|

3 G H

3 G H

}

!

Let

v

` b

g c e

r

X

r

i

then v` b

g c is a computable constant. Obviously we have

|

3 G H

3 G H

}

|

e

r

X

r

|

3 G H

3 G H

}

|

e v

` b

g c

l

3 G H

3 G H

n

`

r

e v

`b

g c

l~

~

e v

` b

g c

l n

(1)

and

l

|

3 G H

}

!

e

ln

(2)

From (1) and (2) we have

r

`

b

X c e

n

v

` b

g c

l n

l n


45/178

A number theoretic function and its mean value 35

e

v

` b

g c

0 l n

This completes the proof of the theorem. As to v` b

g c , it is easy to show that

v

`b

c e

r

r

e

i

and

v

` b

g c e

r

b

X

c

r

e

r

X

r

r

X

r

r

X

r

r

r

e v

` b

g c

v

` b

g c

v

` b

c

v

` b

c i

so we have

v

` b

g c e

n

v

` b

g c

v

` b

c

v

` b

c

From this formula, we can easily compute the first severalv

` b

g c:

v

` b

c e

b

c j

i v

` b

c e

j

b

c

i v

` b

c e

-

j

b

c

`

i

Then use the Theorem, we can get the Corollary.

References


House, 1993.






46/178


47/178

ON THE PRIMITIVE NUMBERS OF POWER

AND

ITS TRIANGLE INEQUALITY

Ding LipingDepartment of Mathematics, Northwest University, Xian, Shaanxi, P.R.China

dingding [email protected]

Abstract The main purpose of this paper is using the elementary method to study the

properties of

, and give a triangle inequality for it.

Keywords: Primitive numbers; Arithmetical property; triangle inequality.

W

1. Introduction

Let be a prime, X be any fixed positive integer,Y ` b

X c denote the smallest

positive integer such thatY ` b

X c uis divisible by r . For example,

Y

b

c e ,

Y

b

c e

,Y

b

c e

,Y

b -

c e

,Y

b

c e

,

. In problem 49 of book

[1], Professor F. Smarandache asks us to study the properties of the sequenceY ` b

X c. About this problem, some asymptotic properties of this sequence have

been studied by many scholar. In this paper, we use the elementary methods to

study the arithmetical properties ofY ` b

X c, and give a triangle inequality for it.

That is, we shall prove the following:

Theorem 1. Let be an odd prime, g be positive integer. Then we have

the triangle inequality

Y `

~

g

Y ` b

g

c

Theorem 2. There are infinite integersg

b

p e i i i csatisfying

Y `

~

g

e

Y ` b

g

c

W


In this section, we complete the proof of the theorems. First we prove the-

orem 1. From the definition ofY ` b

X c, we know that

t

Y ` b

g

c u, t


48/178


Y ` b

g c u(

p e ). From this we can easily obtain:

e

t

Y ` b

g

c u

Y ` b

g c u

t

b Y ` b

g

c

Y ` b

g c c u (1)

But from the definition ofY ` b

X c, we know that

Y ` b

X c uis the smallest positive

integer that is divisible by r . That is

s

t

Y`

b

g

g

c u

(2)

From (1), (2) we can immediately getY ` b

g

g c

Y ` b

g

c

Y ` b

g c

Now the theorem 1 follows from this inequality and the induction.

Next we complete the proof of theorem 2. For any positive integersg

withg

e g

(

p i

), we let }

e

}

b

i X csatisfy

|

X u. Then

}

e

}

b

i X c e

X

For convenient, we let

e

`

E

`

. Since

e

s

j

e

e

So we haveY

`b

c e

i p e i i i (3)

On the other hand,

jk

k

k

k

k

l

mn

n

n

n

n

e

e

So

Y `

~

e

(4)

Combining (3) and (4) we may immediately obtain

Y `

~

e

Y ` b

c


References

[1] Jozsef Sandor, On an generalization of the Smarandache function, Note

Numb.Th.Diser.Math. 5(1999),41-51

[2] Tom M A. Introduction to Analytic Number Theory. New York, 1976.


49/178

THE ADDITIVE ANALOGUE OF SMARANDACHE

SIMPLE FUNCTION

Zhu Minhui1. Math and Phys Dept, XAUEST, Xian, Shaanxi, P.R.China

2. Department of Mathematics, Northwest University, Xian, Shaanxi, [email protected]

Abstract The main purpose of this paper is to study the asymptotic properties of

,

and give two interesting asymptotic formulae for it.

Keywords: Smarandache-simple function; Additive Analogue; Asymptotic formula.

W

1. Introduction and results

For any positiveX

, the Smarandache functionY b

X c

is defined as the smallest

g

, where X t g u . For a fixed prime , the Smarandache-simple functionY ` b

X c is defined as the smallest g

, where

r

t

g u . In reference [2],Jozsef Sandor introduced the additive analogue of the Smarandache-simple

functionY ` b

cas follows:

Y`

b

c e

g

g u i

b

i c i

andY

`

b

c e

7 9

g

g u

i # i c i

which is defined on a subset of real numbers. It is clear thatY ` b

c e gif

b b

g c u i g u$ for

g (for

g e it is not defined, as 0!=1! ), therefore

this function is defined for

. About the properties ofY b

X c

, many people

had studied it before (See [2], [3]). But for the asymptotic formula ofY ` b

c , it

seems that no one have studied it before. The main purpose of this paper is tostudy the asymptotic properties of

Y ` b

c, and obtain an interesting asymptotic

formula for it. That is, we shall prove the following:

Theorem 1. For any real number , we have the asymptotic formula

Y ` b

c e

l

j

Obviously, we have

Y ` b

c e

Y

`

b

c

i p q

b

g u i

b

g

c u c

b

g c

Y

`

b

c i p q e

b

g

c u

b

g c


50/178


Therefore, we can easily get the following:

Theorem 2. For any real number , we have the asymptotic formula

Y

`

b

c e

l

j

W

2. Proof of the theorem

In this section, we complete the proof of the theorem 1. In fact, from the

definition ofY ` b

c, we have

b

g c u

g uand taking the logistic

computation in the two sides of the inequality, we get

p

p (1)

Then using the Eulers summation formula we have

p e

x

b

#

$c

b

c }

x

e g

g g

l

b

g c (2)

and

p e

x

b

#

$c

b

c x

e g

g g

l

b

g c (3)

Combining (1),(2) and (3), we can easily deduce that

e g

g g

l

b

g c (4)

So

g e

g

l

b

c (5)

Similarly, we continue taking logistic computation in two sides of (5), then we

also have

g e

l

b

g c (6)

and

g e

l

b

c (7)


51/178

The additive analogue of Smarandache simple function 41

Hence, by (5), (6) and (7) we have

Y ` b

c e

l

b

g c

l

b

c

e

l

b

g c

b

l

b

g c c

e

l

j


References


House, 1993.

[2] Jozsef Sandor. On additive analogue of certain arithmetic functions,

Smaramche Notes Journal 2004, 14: 128-132.

[3] Mark Farris and Patrick Mitchell, Bounding the smarandache functoin,

Smaramche Notes Journal 2002, 13: 37-42

[4] Tom M.Apostol, Introduction to Analytic Number Theory, Springer-

Verlag, New York,1976.


52/178


53/178

ON THE

-POWER COMPLEMENT SEQUENCE

Yao WeiliResearch Center for Basic Science, Xian Jiaotong University, Xian, Shaanxi, P.R.China

[email protected]

Abstract The main purpose of this paper is using analytic method to study the asymp-

totic properties of -power complement sequence, and give several interesting

asymptotic formulae.

Keywords: -power complement sequence; asymptotic formula; mean value.

W

1. Introdution


, let

b

X cdenotes

-power complement

sequence. That is,

b

X c denotes the smallest integer such that X

b

X c be a

perfect

-power. In problem 29 of reference [1], professor F.Smarandache

asked us to study the properties of this sequence. About this problem, somepeople had studied it before, see references [4]and [5]. The main purpose

of this paper is using the analytic method to study the asymptotic properties

of

-power complement sequence, and obtain several interesting asymptotic

formulae. That is, we shall prove the following :

Theorem. Let xb

X c denote the Dirichlet divisor function, then for any real

number


r

x

b

X

b

X c c e

b

c

l

b

Q

c i

where

,

,

,

are computable constants, is any fixed positive num-

ber.

From this theorem, we may immediately deduce the following

Corollary 1. Letv

b

X cbe the square complement sequence, then for any

real number

, we have

r

x

b

X v

b

X c c e

b

j

c

l

b

Q

c i

where

,

,



54/178


Corollary 2. Let

b

X c be the cubic complement sequence, then for any real

number , we have

r

x

b

X

b

X c c e

b

j

c

l

b

Q

c i

where

,

,

and


W


In this section, we shall complete the proof of the Theorem. Let

q

b

c e

r

x

b

X

b

X c c

X

From the definition of

b

X c, the properties of the divisor function and the

Euler product formula [2], we have

q

b

c e

`

~

x

b

b

c c

x

b

j

b

j

c c

j

e

`

~

x

b

c

x

b

c

x

b

j

c

x

b

j

c

j

e

`

j

e

`

`

p

p

`

`

p

`

e

b

c

`

b

c

e

b

c

b

c

`

b

c

b

c

j

n

b

c

i

where

b

cis the Riemann Zeta function. Obviously, we have

t

x

b

X

b

X c c

t

X i

r

x

b

X

b

X c c

X

i

where is the real part of

. Therefore by Perron formula [3]

r

x

b

X

b

X c c

X


55/178

On the -power complement sequence 45

e

p

q

b

c

x

l~

b

c

l

b

c

i

l

b

c

i

i

where

be the nearest integer to

,

e

t

t . Taking

e

,

e

j

,

e ,

b

c e ,

b

c e

, then we have

r

x

b

X

b

X c c e

p

b

c

b

c

b

c

x

l

b

Q

c i

where

b

c e

`

b

c

b

c

j

n

b

c

To estimate the main term

p

b

c

b

c

b

c

x i

we move the integer line from e

p

to e

p

, then the function

b

c

b

c

b

c

have one order pole point at e with residue

u

b

c

b

c

b

c

b

c

e

u

n

n

b

c

b

c

b

c

b

c

n

n

b

c

b

c

b

c

b

c

n

b

c

b

c

b

c

b

c

e

b

c i

where

i

i i

are computable constants. So we can obtain

p

~

Q

Q

Q

Q

b

c

b

c

b

c x

e

b

c


56/178


Note that the estimates

p

Q

b

c

b

c

b

c x

P

Q

i

p

Q

b

c

b

c

b

c x

P

Q

i

and

p

Q

Q

b

c

b

c

b

c x

P

Q

Therefore we get

r

x

b

X

b

X c c e

b

c

l

b

Q

c

This completes the proof of the Theorem .

References

[1]. F. Smarandache. Only Problems, Not solutions. Chicago: Xiquan

Publishing House, 1993.

[2]. Apostol T M. Introduction to Analytic Number Theory. New York:

Springer- Verlag, 1976.[3]. Pan Chengdong and Pan Chengbiao. Foundation of Analytic Number

Theory. Beijing: Science press, 1997.

[4]. Zhu Weiyi. On the -power complement and -power free number

sequence. Smarandache Notions Journal, 2004, 14: 66-69.

[5]. Liu Hongyan and Lou Yuanbing. A Note on the 29-th Smarandaches

problem. Smarandache Notions Journal, 2004, 14: 156-158.


57/178

ON THE INFERIOR AND SUPERIOR FACTORIAL

PART SEQUENCES

Li JieDepartment of Mathematics, Northwest University, Xian, Shaanxi, P.R.China

[email protected]

Abstract The main purpose of this paper is using the elementary method to study the con-

vergent property of an infinite series involving the inferior and superior factorial

part sequences, and give an sufficient condition of the convergent property of the

series.

Keywords: Inferior factorial part; Superior factorial part; Infinite series.

W

1. Introduction


, the inferior factorial part denoted byv

b

X cis the

largest factorial less than or equal to X . It is the sequence: i i i i i

i

i

i

i

i

i

i

i

i

i

i

i

i

i

i

i

i

i

-

i

-

i

-

i

-

i

-

i

-

i

-

i . On the other

hand, the superior factorial part denoted by

b

X cis the smallest factorial greater

than or equal toX

. It is the sequence as follows: i i

i

i

i

i

-

i

-

i

-

i

-

i

-

i

-

i

-

i

-

i

-

i

-

i

-

i

-

i

-

i

-

i

-

i

-

i

-

i

-

i

i

i

i In the

42th problem (see [1]) of his famous book l Xi

i

g i

i

p

X ,

Professor F.Smarandache asked us to study these sequences. About this prob-

lem, it seems no one had studied it before. In this paper, we studied two infinite

series involving vb

X c and

b

X c as follows:

e

r

v

|

b

X c

i

Y

e

r

|

b

X c

i

and given some sufficient conditions of the convergent property of them. That

is, we shall prove the following

Theorem. Let } be any positive real number. Then the infinite series

andY

are convergent if }

, divergent if }

.

Especially, when }e

, we have the following

Corollary. We have the identity

r

v j

b

X c

e


58/178


W

2. Proof of the theorem

In this section, we will complete the proof of the theorem. Letv

b

X c e g u. It

is easy to see that if g u

X

b

g

c u , then vb

X c e g u . So the same number

g urepeated

b

g

c u g utimes in the sequence

v

b

X c

b

X e i i c.

Hence, we can write

e

r

v

|

b

X c

e

b

g

c u g u

b

g u c

|

e

g g u

b

g u c

|

e

g

b

g u c

|

It is clear that if }

then

is convergent, if }

then

is divergent. Using

the same method, we can also get the sufficient condition of the convergent

property ofY

. Especially, when }e

, from the knowledge of mathematical

analysis (see [2]), we have

r

v j

b

X c

e

b

g c u

e

g u

e

This completes the proof of the theorem.

References

[1] F.Smarandache. Only problems, Not solutions, XiQuan Publishing House,

Chicago, 1993, pp36.

[2] T.M.Apostol. Mathematical analysis, 2nd ed., Addison Wesley Publish-

ing Co., Reading, Mass, 1974.


59/178

A NUMBER THEORETIC FUNCTION AND ITS MEAN

VALUE

Gao NanSchool of Sciences, Department of Mathematics, Xian Shiyou University, Xian, Shaanxi, P.R.China

gaonan 0 0 [email protected]

Abstract Let

and are primes,

denotes the largest exponent of power which

divides

. In this paper, we study the properties of this sequence e

$ f

, and

give an interesting asymptotic formula for

$

e

$ f

.

Keywords: Asymptotic formula; Largest exponent; Mean value.

W

1. Introduction

Let and

are two primes,

b

X c

denotes the largest exponent of power

which dividesX

. It is obvious that e

if

dividesX

but

does not.

In problem 68 of [1], Professor F.Smarandache asked us to study the propertiesof the sequence

b

X c. In this paper, we use elementary methods to study the

asymptotic properties of the mean value

r

r , where is a prime such

that , and give an interesting asymptotic formula for it. That is, we will

prove the following :

Theorem. Let and are primes with . Then for any real number


r

r

e

`

l n

Q

i

if

`

`

`

n

`

`

`

b

c

`

j

`

l n

Q

i if e

where

is any fixed positive number, is the Euler constant.

W


In this section, we will complete the proof of the theorem. Let

q

b

c e

r

r

X


60/178



, it is clear that

b

X cis a addtive function. So

r

is a multiplicative function. Then from the definition of

b

X cand the Euler

product formula (See Theorem 11.6 of [3]), we can write

q

b

c e

r

r

X

e

`

Q

~

`

E

Q

e

`

Q

~

`

Q

`

Q

j

e

~

j

j

`

Q

j

e

b

c

By Perron formula (See reference [2]), let

e

,

e ,

e

j . Then we

have

r

r

e

p

j

j

b

c

b

c

x

l

b

j

c i

where

b

c e

`

and is any fixed positive number.

Now we estimate the main term

p

j

j

b

c

b

c

x i

we move the integral line from

p

to

p

.

If

, then function

b

c

b

c

have a simple pole point at e

, so we have

p

~

j

j

j

j

j

j

j

j

b

c

b

c

x e

b

c

Taking

e

, we have

p

~

Q

j

j

Q

b

c

b

c

x

P

j

Q

b

p

c

b

c

j

x

P

j

e

Q


61/178

A number theoretic function and its mean value 51

And we can easy get the estimate

p

Q

Q

b

c

b

c

x

P

b

p

c

b

c

Q

x

P

Q

Note that

b

c e

i

we may immediately obtain the asymptotic formula

r

r

e

l

b

Q

c i

if

.

If e

, then the function

b

c

b

c

have a second order pole point at e

. Let Res n

b

c

b

c

denotes its

residue, so we have

b

c

b

c

e

b

c

b

c

j

e

~

b

c

b

c

b

c

b

c

b

b

c

b

c c

Noting that

b

c

e

i

b

c

b

c e

and

b

b

c

b

c c e i

we may immediately get

b

c

b

c

e

b

c

So we have the asymptotic formula

r

r

e

b

c

b

c

l

b

Q

c i

if e

. This completes the proof of Theorem.


62/178


References


House, 1993.

[2] Pan Chengdong and Pan Chengbiao. Elements of the analytic number

Theory. Beijing: Science Press, 1991, pp 98.




63/178

ON THE GENERALIZED CONSTRUCTIVE SET

Gou suDepartment of Mathematics and Physics, Xian Institute of Posts and Telecoms, Xian, Shaanxi,

P.R.China

[email protected]

Abstract In this paper, we use the elementary methods to study the convergent properties

of the series

I

$ Q

8

)

$

where)

$

is a generalized constructive number, and is any positive number.

Keywords: Generalized constructive set; Series; Convergent properties.

W

1. Introduction

The generalized constructive setY

is defined as: numbers formed by digits

x

i x

j

i i x

only, allx

being different each other,

g

. That is to

say,

(1) x

i x

j

i i x

belong toY

;

(2) Ifv i

belong toY

, thenv

belongs toY

too;

(3) Only elements obtained by rules (1) and (2) applied a finite number of

times belong toY

.

For example, the constructive set (of digits 1, 2) is : i i i i i i i

i i i i i i i i i i And the constructive

set (of digits 1, 2, 3) is : i i i i i i i i i i i i i i i

i i i i i i i i i In problem 6, 7 and 8 of ref-

erence [1], Professor F.Smarandache asked us to study the properties of this

sequence. About this problem, it seems that no one had studied it before. For

convenience, we denote this sequence by v

r

. In this paper, we shall use the

elementary methods to study the convergent properties of the series

r

v

|

r

i

where } is any positive number. That is , we shall prove the following conclu-

sion:


64/178


Theorem 1. The series

r

v

|

r

is convergent if } g , and divergent

if}

g .

Especially, let

Y

j

e

i

and

Y

e

Then we have the following

Theorem 2. The seriesY

j

andY

are convergent, and the estimate

-

Y

j

C

-

v X x

-

C

Y

-

- - -

hold.

W

2. Proof of the Theorems

In this section, we shall complete the proof of the Theorems. First we prove

Theorem 1. Note that for e i i i i

there areg

numbers of

digits in

the generalized constructive sequence, so we have

r

v

|

r

g

b

c

|

e g

g

|

e

g

e

g

|

|

g

i

where we have used the fact that the series

g

|

is convergent only if its common proportion

ithat is }

g. This

completes the proof of the Theorem 1.

Now we come to prove Theorem 2.

Y

j

e


65/178

On the generalized constructive set 5

Documents

Research on Smarandache Problems in Number Theory, edited by Zhang Wenpeng