About the works of Florentin Smarandache have been written a lot of books (he himself wrote dozens of books and articles regarding math, physics, literature, philosophy). Being a globally recognized personality in both mathematics (there are countless functions and concepts that bear his name) and literature, it is natural that the volume of writings about his research is huge. What we try to do with this encyclopedia is to gather together as much as we can both from Smarandache’s mathematical work and the works of many mathematicians around the world inspired by the Smarandache notions. We structured this book using numbered Definitions, Theorems, Conjectures, Notes and Comments, in order to facilitate an easier reading but also to facilitate references to a specific paragraph. We divided the Bibliography in two parts, Writings by Florentin Smarandache (indexed by the name of books and articles) and Writings on Smarandache notions (indexed by the name of authors). We treated, in this book, about 130 Smarandache type sequences, about 50 Smarandache type functions and many solved or open problems of number theory. We also have, at the end of this book, a proposal for a new Smarandache type notion, id est the concept of “a set of Smarandache-Coman divisors of order k of a composite positive integer n with m prime factors”, notion that seems to have promising applications, at a first glance at least in the study of absolute and relative Fermat pseudoprimes, Carmichael numbers and Poulet numbers. This encyclopedia is both for researchers that will have on hand a tool that will help them “navigate” in the universe of Smarandache type notions and for young math enthusiasts: many of them will be attached by this wonderful branch of mathematics, number theory, reading the works of Florentin Smarandache.

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The math encyclopedia of smarandache type notions

The function named in the title of this book is originated from the exiled Romanian mathematician Florentin Smarandache.

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The Smarandache Function

This function is a generalization of the famous Smarandache function S(n). The main purpose of this paper is using the elementary and analytic methods to study the mean value properties of P(n), and give two interesting mean value formulas for it.

A generalization of the Smarandache function

本书主要介绍Smarandache 函数与伪Smarandache 函数、几类Smarandache 序列及其它相关问题的 最新研究进展, 同时还介绍了一些其它数论问题的研 究成果. 本书各部分内容编排相对独立,有兴趣的读者 可以从阅读本书任何一个章节开始, 开拓读者视野，激发读者对Smarandache 相关问题的研究兴趣. This book mainly introduces the latest research progress of Smarandache functions and pseudo-Smarandache functions, several kinds of Smarandache sequences and other related problems. At the same time it also introduces some other study in number theory. This book commits itself to relatively independent content arrangement with each chapter and section, every reader who is interested in this book can read any chapter for the beginning. It could open up readers’ perspective; arouse readers to study Smarandache problems.

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Smarandache 函数 及其相关问题研究 Vol.8 / Research on Smarandache Functions and Other Related Problems, Vol. 8 [in Chinese Language Only]

For any positive integer n, the Smarandache reciprocal function Sc(n) is defined as the largest positive integer m such that y | n! for all integers 1 ≤ y ≤ m, and m + 1 † n!. The main purpose of this paper is using the elementary and analytic methods to study the mean value distribution properties of Sc(n), and give two interesting mean value formulas for it.

On the Smarandache reciprocal function and its mean value

Let n be any positive integer, the Smarandache function S(n) is deflned as S(n) = minfm : njm!g: In this paper, we discussed the solutions of the following equation involving the Smarandache function: S(m1)+S(m2)+¢¢¢+S(mk) = S(m1 +m2 +¢¢¢+mk); and proved that the equation has inflnity positive integer solutions.

On the solutions of an equation involving the Smarandache function

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On the solutions of an equation involving the Smarandache function