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Smarandache Functions

Several Smarandache Functions, Prime Numbers, and Constants in number theory are presented below.



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Smarandache Functions
by C. Dumitrescu; V. Seleacu
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The function named in the title of this book is originated from the exiled Romanian mathematician Florentin Smarandache, who has significant contributions not only in mathematics, but also in literature. The Smarandache function may be generalised in various ways, one of these generalisations, the Smarandache function attached to a strong divisibility sequence and particulary to Fibonacci sequence, has a dual property with the strong divisibility.
Smarandache Functions
by Wenpeng Zhang; Ling Li
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The main purpose of this paper is to study the solvability of some equations involving the pseudo Smarandache function Z(n) and the Smarandache reciprocal function Sc(n), and propose some interesting conjectures.
Smarandache Functions
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We note that the most unsolved problems of the world on the same subject are related to the Smarandache Function in the Analytic Number Theory.
Smarandache Functions
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Given a positive integer n, let P(n) denote the largest prime factor of n and S(n) denote the smallest integer m such that n divides m!
Smarandache Functions
by Pedro Melendez
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Another problem related to the Smarandache Function.
Smarandache Functions
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Our aim is to show that certain results from a recent paper can be obtained in a simpler way from a generalization.
Smarandache Functions
by M. Andrei; C. Dumitrescu; V. Seleacu; L. Tutescu; St. Zamfir
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In this paper, some remarks on the Smarandache Function are given.
Smarandache Functions
by Thomas Martin
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We propose a problem related to the Smarandache function and offer a solution.
Smarandache Functions
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We present an algorithm in Lattice C to generate S(n).
In two of my previous papers, namely “An interesting property of the primes congruent to 1 mod 45 and an ideea for a function” respectively “On the sum of three consecutive values of the MC function”, I defined the MC function. In this paper I present new interesting properties of three Smarandache type sequences analyzed through the MC function.
Smarandache Functions
by Florian Luca
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For any positive integer n, let S(n) denotes the Smarandache function, then S(n) is defined the smallest m belonging to N+, where n|m!.
Smarandache Functions
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Being the third in a series on the Smarandache Notions, it is a tribute to the mind of Florentin Smarandache that there seems to be no end to the chain of problems. He is to be commended for contributing so many problems in so many areas. It will be at least decades before most of the problems that he has posed will be resolved. If you found this book interesting, I strongly encourage you to examine the references listed at the end of this book. There is much more there that remains unexplored....
Smarandache Functions
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The Pseudo-Smarandache function has a simple definition: Given any integer n > 0, the value of the Pseudo-Smarandache function is the smallest integer m such that n evenly divides the sum 1 + 2 + 3 + ... -+- m. In this paper, several problems concerning this function will be presented and solved. Most will involve the standard number theory functions such as Euler's phi function and the sum of divisors function.
Smarandache Functions
by Marcela Popescu; Paul Popescu
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In our paper we prove that the Smarandache function does not verify the Lipschitz condition.
It is always difficult to talk about arithmetic, because those who do not know what is about, nor do they understand in few sentences, no matter how inspired these might be, and those who know what is about, do no need to be told what is about. Arithmetic is that branch of mathematics that you keep it in your soul and in your mind, not in your suitcase or laptop. Part One of this book of collected papers aims to show new applications of Smarandache function in the study of some well known...
Smarandache Functions
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My intention is to prove that there exists series of arbitrary finite length with the properties described by J. Rodriguez.
Smarandache Functions
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The main purpose of this paper is using the elementary method to study the property of the Smarandache function, and give an interesting result.
Smarandache Functions
by I. Balacenoiu; V. Seleacu
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This paper discusses some properties of the Smarandache Functions of the Type I.
In this paper I define a new type of pairs of primes, id est the Smarandache-Germain pairs of primes, notion related to Sophie Germain primes and also to Smarandache function, and I conjecture that for all pairs of Sophie Germain primes but a definable set of them there exist corespondent pairs of Smarandache-Germain primes. I also make a conjecture that attributes to the set of Sophie Germain primes but a definable subset of them a corespondent set of smaller primes, id est Coman-Germain...
Smarandache Functions
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This paper proposes some remarks concerning the distribution of the Smarandache Function.
Smarandache Functions
by Ion Balacenoiu
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Using certain results on standardised structures, three kinds of Smarandache functions are defined and are etablished some compatibility relations between these functions.
This study is an extension of work done by Charles Ashbacher. Iteration results have been re-defined in terms of invariants and loops. Further empirical studies and analysis of results have helped throw light on a few intriguing questions.
Smarandache Functions
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This paper treats some new inequalities and limits for the Smarandache function.
Smarandache Functions
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Let S(n) be the smallest integer k so that nlk!. This is known as the Smarandache function and has been studied by many authors. If P( n) denotes the largest prime factor of n, it is clear that S(n)≥P(n).
The main purpose of this paper is to introduce some new unsolved problems involving the Smarandache function and the related functions.
S(n) is an even function. That is S(n)= S(-n) since if (S(n))! is divisible by n it is also divisible by -n.
Smarandache Functions
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The main purpose of this paper is using the elementary method to study the properties of Ut(n), and obtain some interesting identities involving function Ut(n).
Smarandache Functions
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this paper proposes an important formula to calculate the number of primes less than x.
Smarandache Functions
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Computer programs for this study were written in UBASIC ver. 8.77. Extensive use was made of NXTPRM(x) and PRMDIV(n) which are very convenient although they also set an upper limit for the search routines designed in the main program.
Smarandache Functions
by A.W. Vyawahare; K. M. Purohit
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This new paper defines a new function K(n) where n belings to N, which is a slight modification of Z(n) by adding a smallest natural number k. Hence this function is "Near Pseudo Smarandache Function (NPSF)". Some properties of K(n) are presented here, separately, according to as n is even or odd. A continued fraction consisting NPSF is shown to be convergent. Finally some properties of Kl (n) are also obtained.
Smarandache Functions
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In this paper, some iterations of this function on palindromes that yield palindromes are demonstrated.
Smarandache Functions
by A. Stuparu; D. W. Sharpe
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We prove that the equation S(x) = p, where p is a given prime number has just D((p-1)!) solutions, all of them in between p and p!.
Smarandache Functions
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The Smarandache function satisfies certain elementary inequalities which have importance in the deduction of properties of this (or related) functions.
Smarandache Functions
by Jim Duncan
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The ratio of the number of ones to the number of zeros appears to be approximately 1 for large values of k.
Smarandache Functions
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The main purpose of this paper is using the elementary method to study the estimate problem of S (Fn), and give a sharper lower bound estimate for it.
Smarandache Functions
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This paper proposes some elementary algebraic considerations inspired by the Smarandache Function.
Smarandache Functions
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This topic is certain to be revisited in the near future and the lack of space available here will certainly be remedied on that occasion.
Smarandache Functions
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A program SMARAND has been designed to generate S(n) up to a preset limit N (N up to 1000000 has been used in some applications).
Smarandache Functions
by David Gorski
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The Pseudo-Smarandache Function is part of number theory. The function comes from the Smarandache Function. The Pseudo-Smarandache Function is represented by Z(n) where n represents any natural number.
Smarandache Functions
by Ken Tauscher
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Let N be a positif integer. Let η be the function that associates to any non-null integer P the smallest number Q such find the minimwn value of K from which η(R) > N for any R > K.
In order to make students from the American competions to learn and understand better this notion, used in many east - european national mathematical competions, the author: calculates it for some small numbers, establishes a few proprieties of it, and involves it in relations with other famous functions in the number theory.
Smarandache Functions
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New references concerninig Smarandache function.
Smarandache Functions
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The Smarandache function is an integer function, S, of an integer variable, n. S is the smallest integer such that S! is divisible by n.
Smarandache Functions
by Charles Ashbacher
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In this paper we shall investigate some aspects involving Smarandache function.
Smarandache Functions
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This paper proves the existence of an infinite family of pairs of dissimilar Pythagorean triangles that are pseudo Smarandache related.
Smarandache Functions
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This paper proposes some open questions for the Smarandache Function.
Smarandache Functions
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This paper deals with palindromic numbers and iterations of the Pseudo-Smarandache Function.
Studying the two well known recurrent relations with the exceptional property that they generate only values which are equal to 1 or are primes, id est the formula which belongs to Eric Rowland and the one that belongs to Benoit Cloitre, I managed to discover a formula based on Smarandache function, from the same family of recurrent relations, which, instead to give a prime value for any input, seems to give the same value, 2, if and only if the value of the input is a prime. I name this...
Smarandache Functions
by Steven R. Finch
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This paper extends earlier work on the average value of the Smarandache function S(n) and is based on a recent asymptotic result.
Smarandache Functions
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We have constructed a function n which associates to each non-null integer m the smallest positive n such that n! is a multiple of m.
The main purpose of this paper is using the elementary methods to study the hybrid mean value of the Smarandache function S(n) and the Mangoldt function Λ(n), and prove an interesting hybrid mean value formula for S(n)Λ(n).
Recently I. Cojocaru and S. Cojocaru have proved a certian irrationality regarding the Smarandache Function. The author of this note showed that this is a consequence of an old irrationality criteria (which will be used here once again), and proved a result implying the given irrationality.
Smarandache Functions
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This paper shows a way of alculating the Smarandache Function without factorising.
In this paper we give a survey on recent results on Smarandache function.
Smarandache Functions
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Mike Mudge pays a return visit to the Florentin Smarandache Function.
Smarandache Functions
by Zhang Wenpeng; Xu Zhefeng
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The main purpose of this paper is using the elementary method to study the mean value properties of the Smarandache function, and give an interesting asymptotic formula.
Smarandache Functions
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The main purpose of this paper is using the elementary method to study the number of the solutions of a given congruent equation and give its all prime number solutions.
Smarandache Functions
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For any positive integer n, let S(n) denote the Smarandache function of n. In this paper, we prove that S(mn)≤S(m)≤S(n).
Smarandache Functions
by C. Dumitrescu; C. Rocsoreanu
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The Smarandache function S : N*-N* is defined by the condition that S(n) is the smallest integer m such that m! is divisible by n.
The main purpose of this paper is using the elementary methods to study the mean value properties of the Pseudo-Smarandache-Squarefree function and Smarandache function, and give two sharper asymptotic formulas for it.
Smarandache Functions
by Vasile Seleacu; Narcisa Virlan
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In this paper is studied the limit of a sequence.
Smarandache Functions
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The main purpose of this paper is using the elementary methods to study the mean value properties of p(n)/Z(n), and give a sharper asymptotic formula for it, where p(n) denotes the smallest prime divisor of n.
Smarandache Functions
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The main purpose of this paper is using the elementary method to study the properties of the Pseudo Smarandache function Z(n), and solve two conjectures posed by Kenichiro Kashihara.
Smarandache Functions
by Su Gou; Jianghua Li
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The main purpose of this paper is using the elementary method to study the properties of the Pseudo-Smarandache function Z(n), and proved the following two conclusions: The equation Z(n) = Z(n+ 1) has no positive integer solutions; For any given positive integer M, there exists an integer s such that the absolute value of Z(s) − Z(s + 1) is greater than M.
Smarandache Functions
by Constantin Dumitrescu
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Sn is defined as the sequence obtained through the concatenation of the first n odd numbers (the n-th term of the sequence is formed through the concatenation of the odd numbers from 1 to 2*n – 1).
Smarandache Functions
by T. Yau
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This paper proposes a problem of maximum.
Smarandache Functions
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In this paper, we use the elementary method to study this problem, and prove that for any integer n≥1, the inequality has infinite group positive integer solutions (x1, x2, … , xn).
Smarandache Functions
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This is a paper about the Pseudo-Smarandache Function.
Smarandache Functions
by I. Balacenoiu; V. Seleacu
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This paper recapitulates the sientific papers dedicated to the Smarandache function.
Smarandache Functions
by Sebastian Martin Ruiz
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Smarandache Function is defined as followed: S(m) = The smallest positive integer so that S(m)! is divisible by m.
Smarandache Functions
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All the values of S(n) for n≤32000, conveniently chosen in order to use short integers only, have been sorted.
Smarandache Functions
by Pedro Melendez
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A new problem related to the Smarandache Function.
Smarandache Functions
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The study of infinite series involving Smarandache function is one of the most interesting aspects of analysis.
Smarandache Functions
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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.
The Smarandache's universe is undoubtedly very fascinating and is halfway between the number theory and the recreational mathematics. Even though sometime this universe has a very simple structure from number theory standpoint, it doesn't cease to be deeply mysterious and interesting. This book, following the Smarandache spirit, presents new Smarandache functions, new conjectures, solved/unsolved problems, new Smarandache type sequences and new Smarandache Notions in number theory.