RESEARCH ON NUMBER THEORY AND SMARANDACHE NOTIONS （PROCEEDINGS OF THE SIXTH INTERNATIONAL CONFERENCE ON NUMBER THEORY AND SMARANDACHE NOTIONS）

2010·3.9 MB·English

by  Zhang, Wenpeng.

RESEARCH ON NUMBER THEORY AND
SMARANDACHE NOTIONS

（PROCEEDINGS OF THE SIXTH INTERNATIONAL
CONFERENCE ON NUMBER THEORY AND
SMARANDACHE NOTIONS）

Edited by
ZHANG WENPENG
Department of Mathematics
Northwest University
Xi'an, P.R.China

Hexis
2010
i
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Peer Reviewer:
Shigreru Kanemitsu, Graduate School of Advanced Technology of Kinki, 11-6
Kayanomori, Iizuka 820-8555, Japan.
Zhai Wenguang, Department of Mathematics, Shandong Teachers’ University, Jinan,
Shandong, P.R.China.
Guo Jinbao, College of Mathematics and Computer Science, Yanan University,
Shaanxi, P.R.China.

Copyright 2010 By Hexis and Zhang Wenpeng, and all authors for their own articles

Many books can be downloaded from the following E-Libraray of Science:

http: // www. gallup.unm.edu /~smarandache/eBooks-otherformats.htm

ISBN: 9781599731278

Standard Address Number : 297-5092
Printed in the United States of America

ii
Preface

This Book is devoted to the proceedings of the Sixth International Conference
on Number Theory and Smarandache Notions held in Tianshui during April 24-25,
2010. The organizers were myself and Professor Wangsheng He from Tianshui
Normal University. The conference was supported by Tianshui Normal University
and there were more than 100 participants. We had one foreign guest, Professor
K.Chakraborty from India. The conference was a great success and will give a
strong impact on the development of number theory in general and Smarandache
Notions in particular. We hope this will become a tradition in our country and will
continue to grow. And indeed we are planning to organize the seventh conference
in coming March which will be held in Weinan, a beautiful city of shaanxi.
In the volume we assemble not only those papers which were presented at the
conference but also those papers which were submitted later and are concerned
with the Smarandache type problems or other mathematical problems.
There are a few papers which are not directly related to but should fall within
the scope of Smarandache type problems. They are 1. A. K. S. Chandra Sekhar Rao,
On Smarandache Semigroups; 2. X. Pan and Y. Shao, A Note on Smarandache
non-associative rings; 3. Jiangmin Gu, A arithmetical function mean value of
binary; etc.
Other papers are concerned with the number-theoretic Smarandache problems
and will enrich the already rich stock of results on them.
Readers can learn various techniques used in number theory and will get
familiar with the beautiful identities and sharp asymptotic formulas obtained in the
volume.
Researchers can download books on the Smarandache notions from the
following open source Digital Library of Science:
www.gallup.unm.edu/~smarandache/eBooks-otherformats.htm.

Wenpeng Zhang
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Contents
X. Li and X. Wu : The Smarandache sums of products for E(n,r) and O(n,r) 1
T. G. Ja´ıy´eo.l´a : Smarandache isotopy of second Smarandache Bol loops 8
J. Gu : A arithmetical function mean value of binary 20
W. Yang : The adjoint semiring part of IS-algebras 28
X. Yuan : On the Smarandache double factorial function 32
A. K. S. C. S. Rao : On Smarandache Semigroups 38
S. Ren, etc. : Merrifield-Simmons index of zig-zag tree-type hexagonal systems 43
X. Wu and X. Li : An equation involving function Sc(n) and Z∗(n) 52
M. Yang : The conjecture of Wenpeng Zhang with respect to the Smarandache
3n-digital sequence 57
X. Ren and N. Feng : A structure theorem of left U-rpp semigroups 62
X. Lu : On the Smarandache 5n-digital sequence 68
Y. Shao and L. Zhai : Amenable partial orders on a locally inverse semigroup 74
L. Huan : On the generalization of the Smarandache’s Cevians Theorem (II) 81
X. Ren and J. Wang : The translational hull of strongly right Ehresmann
semigroups 86
X. Pan and Y. Shao : A Note on Smarandache non-associative rings 93
M. Xiao : An equation involving Z∗(n) and SL(n) 97
W. Huang : On the mean value of Smarandache prime part P (n) and p (n) 100
p p
J. Tian and Y. Shao : A representation of cyclic commutative asynchronous
automata 105
H. He and Q. Zhao : Partial Lagrangian and conservation laws for the perturbed
Boussinesq partial differential equation 111
J. Hu : Some formulate for the Fibonacci and Lucas numbers 119
P. Fan : Some identities involving the classical Catalan Numbers 128
Q. Zhao, etc. : Hosoya index of zig-zag tree-type hexagonal systems 132
iv
The Sixth International Conference on Number Theory
and Smarandache Notions

The opening ceremony of the conference is occurred in Tianshui Normal University
(http://www.tsnc.edu.cn).

Professor Xinke Yang

v
Professor Wenpeng Zhang

Professor Kalyan Chakraborty

vi
Professor Hailong Li

Professor Wansheng He

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viii
The Smarandache sums of products for
E(n,r) and O(n,r)
Xiaoyan Li † and Xin Wu ‡
Department of Mathematics, Northwest University, Xi’an, Shaanxi, P.R.China
Abstract The main purpose of this paper is using the elementary methods to study the
properties of the function E(n,r) and O(n,r), and get two calculating formulaes for them.
Keywords The Smarandache sums of products, binomial theorem.
§1. Introduction
Thispaperdealswiththesumsofproductsoffirstnevenandoddnaturalnumbers,taken
r at a time. Many interesting results about these two functions are obtained. For example,
Mr. Ramasubramanian [1] and Anant W. Vyawahare [3] have already made some work in this
direction. This paper is an extension of their work.
Definition. For any positive integer n and r, E(n,r) are the sums of products of first
n even natural numbers, taken r at a time, r ≤ n. O(n,r) are the sums of products of first n
odd natural numbers, taken r at a time, r ≤n, they are also without repeatition.
For example:
E(4,1)=2+4+6+8=20,
E(4,2)=2·4+2·6+2·8+4·6+4·8+6·8=140,
E(4,3)=2·4·6+2·4·8+4·6·8+2·6·8=400,
E(4,4)=2·4·6·8=384,
O(4,1)=3+5+7+9=24,
O(4,2)=3·5+3·7+3·9+5·7+5·9+7·9=206,
O(4,3)=3·5·7+3·5·9+5·7·9+3·7·9=744,
O(4,4)=3·5·7·9=945.
We assume that E(n,0)=O(n,0)=1.
About the properties of functions E(n,r) and O(n,r), we can obtain some interesting
conclusions from their definitions. Following are some elementary properties of E(n,r) and
O(n,r):
1. E(n,n)=2nE(n−1,n−1),
2. O(n,n)=(2n+1)O(n−1,n−1),
3. E(n,1)=n(n+1),
4. O(n,1)=n(n+2),
5. (p+2)(p+4)(p+6)···(p+2n)=E(n,0)pn+E(n,1)pn−1+E(n,2)pn−2+E(n,3)pn−3+
···+E(n,n−1)p+E(n,n),