Vol. 1, No. 1, 2005 ISSN 1556-6706 SCIENTIA MAGNA Department of Mathematics Northwest University Xi’an, Shaanxi, P. R. China Scientia Magna is published annually in 200-300 pages per volume and 1,000 copies. It is also available in microfilm format and can be ordered (online too) from: Books on Demand ProQuest Information & Learning 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Michigan 48106-1346, USA Tel.: 1-800-521-0600 (Customer Service) URL: http://wwwlib.umi.com/bod/ Scientia Magna is a referred journal: reviewed, indexed, cited by the following journals: "Zentralblatt Für Mathematik" (Germany), "Referativnyi Zhurnal" and "Matematika" (Academia Nauk, Russia), "Mathematical Reviews" (USA), "Computing Review" (USA), Institute for Scientific Information (PA, USA), "Library of Congress Subject Headings" (USA). Printed in the United States of America Price: US$ 69.95 Information for Authors Papers in electronic form are accepted. 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Each author will receive a free copy of the journal. Contributing to Scientia Magna Authors of papers in science (mathematics, physics, philosophy, psychology, sociology, linguistics) should submit manuscripts to the main editor: Prof. Dr. Zhang Wenpeng, Department of Mathematics, Northwest University, Xi’an, Shaanxi, P. R. China, E-mail: [email protected] and another copy to: L. Cuciuc, 200 College Road, UNM-Gallup, NM 87301, USA, E-mail: [email protected]. Associate Editors Dr. W. B. Vasantha Kandasamy, Department of Mathematics, Indian Institute of Technology, IIT Madras, Chennai - 600 036, Tamil Nadu, India. Dr. Larissa Borissova and Dmitri Rabounski, Sirenevi boulevard 69-1-65, Moscow 105484, Russia. Dr. Liu Huaning, Department of Mathematics, Northwest University, Xi’an, Shaanxi, P.R.China, E-mail: [email protected]. Prof. Yi Yuan, Research Center for Basic Science, Xi’an Jiaotong University, Xi’an, Shaanxi, P.R.China, E-mail: [email protected]. Dr. Xu Zhefeng, Department of Mathematics, Northwest University, Xi’an, Shaanxi, P.R.China, E-mail: [email protected]. Dr. Zhang Tianping, College of Mathematics and Information Science, Shaanxi Normal University, Xi’an, Shaanxi, P.R.China, E-mail: [email protected]. Contents On the Smarandache function and square complements 1 Zhang Wenpeng, Xu Zhefeng On the integer part of the k-th root of a positive integer 5 Zhang Tianping, Ma Yuankui Smarandache “Chopped” N N and N + 1N¡1 9 Jason Earls The 57-th Smarandache’s problem II 13 Liu Huaning, Gao Jing Perfect Powers in Smarandache n- Expressions 15 Muneer Jebreel Karama On the m-th power residue of n 25 Li Junzhuang and Zhao Jian Generalization of the divisor products and proper divisor products sequences 29 Liang Fangchi The science of lucky sciences 33 Jon Perry Smarandache Sequence of Unhappy Numbers 37 Muneer Jebreel Karama On m-th power free part of an integer 39 Zhao Xiaopeng and Ren Zhibin On two new arithmetic functions and the k-power complement number sequences 43 Xu Zhefeng Smarandache Replicating Digital Function Numbers 49 Jason Earls On the m-power residues numbers sequence 53 Ma Yuankui, Zhang Tianping vi SCIENTIA MAGNA VOL.1, NO.1 Smarandache Reverse Power Summation Numbers 57 Jason Earls Some Smarandache Identities 59 Muneer Jebreel Karama On the integer part of a positive integer’s k-th root 61 Yang Hai, Fu Ruiqin Smarandache Friendly Cube Numbers 67 Muneer Jebreel Karama Some Expressions of the Smarandache Prime Function 71 Sebastian Martin Ruiz An Improved Algorithm for Calculating the Sum-of-Factorials Function 75 Jon Perry On the Smarandche function and its hybrid mean value 79 Yao Weili On the 83-th Problem of F. Smarandache 83 Gao Nan On Smarandache triple factorial function 89 You Qiying On k-factorials and Smarandacheials 93 Jon Perry A note on Exponential Divisors and Related Arithmetic Functions 97 Jo¶zsef Sa¶ndor Smarandache multiplicative function 103 Liu Yanni, Gao Peng Two formulas for Smarandache LCM ratio sequences 109 Wang Ting The 97-th problem of F.Smarandache 115 Yi Yuan On Two Subsets of Generalized Smarandache Palindromes 119 Jason Earls The Smarandache factorial sequence 123 Zhang Xiaobeng The Smarandache multiplicative function 125 Ma Jinping On Consecutive Values of the Smarandache Function 129 Contents vii Jason Earls On the 82-th Smarandache’s Problem 131 Fu Ruiqin, Yang Hai On a New Class of Smarandache Prime Numbers 135 Jason Earls On the odd sieve sequence 137 Yao Weili On the k-power part residue function 141 Yang Hai, Fu Ruiqin Mean value of the additive analogue of Smarandache function 145 Yi Yuan and Zhang Wenpeng Hybrid mean value on some Smarandache-type multiplicative functions and the Mangoldt function 149 Liu Huaning, Gao Jing On a number set related to the k-free numbers 153 Li Congwei Smarandache Pseudo– Happy numbers 157 Anant W. Vyawahare A number theoretic function and its mean value 163 Ren Ganglian A new function and its mean value 167 Ding Liping On the m-power complement numbers 171 Zhang Xiaobeng On the primitive numbers of power p and its asymptotic property 175 Yi Yuan Mean value of the additive analogue of Smarandache function 179 Zhu Minhui On the generalization of the floor of the square root sequence 183 Yao Weili Mean value of a new arithmetic function 187 Liu Yanni, Gao Peng On the number of numbers with a given digit sum 191 Jon Perry On the mean value of the Smarandache double factorial function 197 Zhu Minhui viii SCIENTIA MAGNA VOL.1, NO.1 On the m-power free part of an integer 203 Liu Yanni, Gao Peng On the mean value of the SCBF function 207 Zhang Xiaobeng ON THE SMARANDACHE FUNCTION AND SQUARE COMPLEMENTS ¤ Zhang Wenpeng Research Center for Basic Science, Xi’an Jiaotong University, Xi’an, Shaanxi, P.R.China [email protected] Xu Zhefeng Department of Mathematics, Northwest University, Xi’an, P.R.China [email protected] Abstract 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. Keywords: Smarandache function; Square complements; Asymtotic formula. x1. Introduction Let n be an positive integer, if a(n) is the smallest integer such that na(n) is a perfect square number, then we call a(n) as the square complements of n. The famous Smarandache function S(n) is defined as following: S(n) = minfm : m 2 N; njm!g: In problem 27 of [1], Professor F. Smarandache let us to study the properties of the square complements. It seems no one know the relation between this sequence and the Smaradache function before. In this paper, we shall study the mean value properties of the Smarandache function acting on the square complements, and give an interesting asymptotic formula for it. That is, we shall prove the following conclusion: ¤This work is supported by the N.S.F.(10271093,60472068) and the P.N.S.F.of P.R.China . 2 SCIENTIA MAGNA VOL.1, NO.1 Theorem. For any real number x ¸ 3, we have the asymptotic formula à ! X ¼2x2 x2 S(a(n)) = + O : 12 ln x 2 n·x ln x x2. Proof of the theorem To complete the proof of the theorem, we need some simple Lemmas. For convenience, we denote the greatest prime divisor of npby p(n). Lemma 1. If n is a square free number or p(n) > n, then S(n) = p(n). Proof. (i) n is a square free number. Let n = p1p2 ¢ ¢ ¢ prp(n), then pijp(n)!; i = 1; 2; ¢ ¢ ¢ ; r: So njp(n)!, butpp(n) y (p(n) ¡ 1)!, so n y (p(n) ¡ 1)!, that is, S(n) = p(n); ®1 ®2 ®r (ii) p(n) > n. Let n = p1 p2 ¢ ¢ ¢ pr p(n), so we have p ®1 ®2 ®r p1 p2 ¢ ¢ ¢ pr < n then ®i pi jp(n)!; i = 1; 2; ¢ ¢ ¢ ; r: So njp(n)!, but p(n) y (p(n) ¡ 1)!, so S(n) = p(n). This proves Lemma 1. Lemma 2. Let p be a prime, then we have the asymptotic formula à ! X x2 x2 p = + O : p 2 ln x ln2 x x·p·x Proof. Let ¼(x) denotes the number of the primes up to x. Noting that µ ¶ x x ¼(x) = + O ; ln x ln2 x from the Abel’s identity [2], we have X p p Z x p = ¼(x)x ¡ ¼( x) x ¡ p ¼(t)dt p x x·p·x à ! x2 1 x2 x2 = ¡ + O ln x 2 ln x ln2 x à ! x2 x2 = + O : 2 ln x ln2 x This proves Lemma 2. Now we prove the theorem.