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Scientia Magna : An International Journal : Volume 3, No. 3, 2007

By Xi'an, Shaanxi

Scientia Magna is published annually in 200-300 pages per volume and 1,000 copies on topics such as mathematics, physics, philosophy, psychology, sociology, and linguistics.

An identity involving the function ep(n) Abstract The main purpose of this paper is to study the relationship between the Riemann zeta-function and an in¯nite series involving the Smarandache function ep(n) by using the elementary method, and give an interesting identity. Keywords Riemann zeta-function, in¯nite series, identity. x1. Introduction and Results Let p be any fixed prime, n be any positive integer, ep(n) denotes the largest exponent of power p in n. That is, ep(n) = m, if pm j n and pm+1 - n. In problem 68 of [1], Professor F.Smarandache asked us to study the properties of the sequence fep(n)g. About the elementary properties of this function, many scholars have studied it (see reference [2]-[7]), and got some useful results. For examples, Liu Yanni [2] studied the mean value properties of ep(bk(n)), where bk(n) denotes the k-th free part of n, and obtained an interesting mean value formula for it. That is, let p be a prime, k be any fixed positive integer, then for any real number x ¸ 1, we have the asymptotic formula.

Table of Contents
F. Ayatollah, etc. : Some faces of Smarandache semigroups' concept in transformation semigroups' approach 1 G. Feng : On the F.Smarandache LCM function 5 N. Quang and P. Tuan : An extension of Davenport's theorem 9 M. Zhu : On the hybrid power mean of the character sums and the general Kloosterman sums 14 H. Yang and R. Fu : An equation involving the square sum of natural numbers and Smarandache primitive function 18 X. Pan and P. Zhang : An identity involving the Smarandache function ep(n) 26 A.A.K. Majumdar : A note on the Smarandache inversion sequence 30 F. Li : Some Dirichlet series involving special sequences 36 Y. Wang : An asymptotic formula of Sk(n!) 40 S. Gao and Z. Shao : A fuzzy relaxed approach for multi-objective transportation problem 44 J. Li : An infinity series involving the Smarandache-type function 52 B.E. Carvajal-Gamez, etc. : On the Lorentz matrix in terms of Infeld-van der Waerden symbols 56 X. Li : On the mean value of the Smarandache LCM function 58 K. Ran and S. Gao : Ishikawa iterative approximation of fixed points for multi-valued Ástrongly pseudo-contract mappings 63 X. Fan : On the divisibility of the Smarandache combinatorial sequence 70 X. Yan : On the Smarandache prime part 74 W. He : The existence of solutions to the Direchlet problem 78 C. Aguilar-Chavez, etc. : On the fifth-kind Chebyshev polynomials 82 S. Luo : Deficient functions of random Dirichlet series of finite order in the half plane 85 J. Li : On the Pseudo-Smarandache-Squarefree function 93 L. Cheng : On the mean value of the Pseudo-Smarandache function 97 W. He : A class of exact solution of the BBM equation 101 C. Tian : An equation involving the two Smarandache LCM dual functions 104 Y. Guo : About Smarandache prime additive complement 108 Y. Lu : Some new problems about the Smarandache function and related problems 110


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