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Research on Number Theory and Smarandache Notions : Proceedings of the Fifth International Conference on Number Theory and Smarandache Notions

By Smarandache, Florentin

Book Id:WPLBN0002828532 Format Type:PDF (eBook) File Size:3.70 mb Reproduction Date:8/6/2013

Smarandache, F. (2013). Research on Number Theory and Smarandache Notions : Proceedings of the Fifth International Conference on Number Theory and Smarandache Notions. Retrieved from http://ebooklibrary.org/

Description
This book contains 23 papers, most of which were written by participants to the fifth International Conference on Number Theory and Smarandache Notions held in Shangluo University, China, in March, 2009. In this Conference, several professors gave a talk on Smarandache Notions and many participants lectured on them both extensively and intensively. All these papers are original and have been refereed. The themes of these papers range from the mean value or hybrid mean value of Smarandache type functions, the mean value of some famous number theroretic functions acting on the Smarandache sequences, to the convergence property of some infinite series involving the Smarandache type sequences.

Summary
This Book is devoted to the proceedings of the fifth International Conference on Number Theory and Smarandache Notions held in Shangluo during March 27-30, 2009. 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. 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.

Excerpt
3. Remarks
Sandor [2] has considered the problem of finding the S-perfect and completely S-perfect numbers, but his proof is not complete. He has proved that the only S-perfect of the form n = p q is n = 6 and there is no S-perfect number of the form n = 2kq where k ¸ 2 is an integer and q is an odd prime. On the other hand, Theorem 2.1 gives all the S-perfect numbers. Again, Sandor only proved that, the only completely S-perfect number of the form n = p2q is n = 28, and all completely S-perfect numbers are given by Theorem 2.2. Theorem 2.1 and Theorem 2.2 ¯nd respectively the S-perfect and completely S-perfect numbers when S(1) = 1. The situation is quite different if one adopts the convention that S(0) = 1. In the latter case, as has been proved by Gronas [3], all completely S-perfect numbers are n = p(prime), 9, 16, 24. All that is known about the S-perfect numbers is that, among the ¯rst 106 numbers, n = 12 is the only S-perfect number (see Ashbacher [4]). In exactly the same way, the Z-perfect and completely Z-perfect numbers may be defined. Thus, given an integer n, 1.

Table of Contents
J. Wang : An equation related to the Smarandache power function 1
X. Lu and J. Hu : On the F.Smarandache 3n-digital sequence 5
B. Cheng : An equation involving the Smarandache double factorial function
and Euler function 8
A. A. K. Majumdar : S-perfect and completely S-perfect numbers 12
B. Zhao and S. Wang: Cyclic dualizing elements in Girard quantales 21
P. Chun and Y. Zhao : On an equation involving the Smarandache function
and the Dirichlet divisor function 27
F. Li and Y. Wang : An equation involving the Euler function and the
Smarandache m-th power residues function 31
H. Gunarto and A. Majumdar : On numerical values of Z(n) 34
K. Nagarajan, etc.: M-graphoidal path covers of a graph 58
H. Liu : On the cubic Gauss sums and its fourth power mean 68
A. A. K. Majumdar : On the dual functions Z¤(n) and S¤(n) 74
S. S. Gupta : Smarandache sequence of Ulam numbers 78
L. Li : A new Smarandache multiplicative function and its arithmetical properties 83
Y. Yang and X. Kang : A predator-prey epidemic model with infected predator 86
R. Fu and H. Yang : An equation involving the Lucas numbers 90
W. Yang : Two rings in IS-algebras 98
W. Zhu : An identity related to Dedekind sums 102
X. Yuan : An equation involving the Smarandache dual function
and Smarandache ceil function 106
Y. Wang : On the quadratic mean value of the Smarandache dual function S¤¤(n) 109
P. Xi and Y. Yi : Distribution of quadratic residues over short intervals 116
Y. Liu : On an equation involving Smarandache power function 120
S. Shang and G. Chen : A new Smarandache multiplicative function and its
mean value formula 123
X. Li : The mean value of the k-th Smarandache dual function 128