Smarandache Notions Journal, Vol. 10

Vol. 10, No. 1-2-3, Spring 1999

ISSN 1084-2810

SMARANDACHE NOTIONS JOURNAL

Number Theory Association of the

UNIVERSITY OF CRAIOVA

The scheme on the first cover represents fraction as a function of n [see p. 45].

a

general

This journal is an extension of the "Smarandache Journal", ISBN 1053-4792, Vol. 1-6, 1990-1995.

continued Function

SNJ is yearly published (in the Spring/Fall - and sometimes in the Summer too) in a 150-250 pages volume, and 800-1000 copies. SNJ is a referred journal. SNJ is reviewed, indexed, citated by the following journals: "Zentralblatt Fuer Mathematik" (Germany), "Referativnyi Zhurnal" and "Matematika" (Academia Nauk, Russia), "Mathematical Reviews" (USA), "Computing Review" (USA), "Libertas Mathematica" (USA), "Octogon" (Romania), "Indian Science Abstracts" (India), "Ulrich's International Periodicals Directory" (USA), "Gale Directory of Publications and Broadcast Media" (USA), "Historia Mathematica" (USA), "Journal of Recreational Mathematics" (USA), "The Mathematical Gazette" (U.K.), "Abstracts of Papers Presented to the American Mathematical Society" (USA), "Personal Computer World" (U.K.), "Mathematical Spectrum" (U.K.), "Bulletin of Pure and Applied Sciences" (India), Institute for Scientific Information (PA, USA), "Library of Congress Subject Headings" (USA). Referents: Y. V. Chebrakov, Department of Mathematics, St.-Petersburg Technical University, Nevsky 3-11, 191186, St.-Petersburg, Russia. Ion Goian, Department of Algebra, Number Theory, and Logic, State University of Kishinev, Republic of Moldova. Vasile Marin, Department of Algebra, Number Theory, and Logic, State University of Kishinev, Republic of Moldova. Florian Luca, Department of Mathematics, Syracuse University, Syracuse, NY 13244-1150, USA. Sebastian Martin Ruiz, Avda. De Regla, 43, Chipiona 11550 (Cadiz), Spain. Sabin T|bârc|, Department of Computer Science, University, BraÕov, Romania. Copyright 1999 by Number Theory Association of the University of Craiova, 13 Al. I. Cuza Street, R-1100, Romania.

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ASSOCIATE EDITORIAL BOARD: Charles Ashbacher, Charles Ashbacher Technologies, Box 294, 119 Northwood Drive, Hiawatha, Iowa 52233, USA. Mihaly Bencze, Str. H|rmanului 6, 2212 S|cele 3, Jud. BraÕov, Romania. Emil Burton, Department of Mathematics, University of ClujNapoca, Romania. Ion Cojocaru & Sorin Cojocaru, Department of Mathematics, University of Craiova, Romania. Constantin Corduneanu, Department of Mathematics, University of Texas at Arlington, USA. Henry Ibstedt, Glimminge 2036, 280 60 Broby, Sweden. Widmer Lamarr, Department of Mathematics, Mesiah College, Grantham, PA 17027, USA. Maohua Le, Department of Mathematics, Zhanjiang Normal College, 29 Cunjin Road, Chikan, Zhanjiang, Guangdong, P.R.China. A.A.K. Majumdar, Department of Mathematics, Jahangirnagar University, Savar, Dhaka 1342, Bangladesh. Mike Mudge, 22 Gors Fach, Pwll-Trap, St Clears, Carmarthenshire, Dyfed SA33 4AQ, United Kingdom. Jean Normand, 3, square Robinson, 92330 Sceaux, France. Stefan Porubsky, Department of Mathematics, Institute of Chemical Technology, Technicka 1905, 16628 Prague 6, Czech Republic. Department of Mathematics, E. R|descu & N. R|descu, University of Craiova, Romania. Carmen RocÕoreanu, Department of Mathematics, University of Craiova, Romania. Francisco Bellot Rosado, Dos de Mayo 16-8E dcha, E-47004 Valladolid, Spain. Jozsef Sandor, Department of Mathematics, BabeÕ-Bolyai University, 3400 Cluj-Napoca, Romania. Y. A. Shmagin, Department of Mathematics, St.-Petersburg Technical University, Nevsky 3-11, 191186, St.-Petersburg, Russia. Vera W. de Spinadel, Centro de Matematica y Diseno MAyDI, Facultad de Arquitectura, Diseno y Urbanismo, Universidad de Buenos Aires, Argentina. Homer B. Tilton, Department of Mathematics, Phisics, and Astronomy, Pima Community College, Tucson, Arizona, USA.

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INDICATION TO AUTHORS Authors of papers concerning any of Smarandache type functions, numbers, sequences, integer algorithms, paradoxes, experimental geometries, etc. are encouraged to submit manuscripts to the Editors: V. Seleacu & I. B|l|cenoiu Department of Mathematics University of Craiova, Romania. The submitted manuscripts may be in the format of remarks, conjectures, solved/unsolved or open new proposed problems, notes, articles, miscellaneous, etc. They must be original work and camera ready [typewritten/computerized, format: 8.5 x 11 inches (. 21,6 x 28 cm)]. They are not returned, hence we advise the authors to keep a copy. The title of the paper should be writting with capital letters. The author's name has to apply in the middle of the line, near the title. References should be mentioned in the text by a number in square brackets and should be listed alphabetically. Current address followed by e-mail address should apply at the end of the paper, after the references. The paper should have at the beginning an up to a half-page abstract, followed by the key words. All manuscripts are subject to anonymous review by two or more independent sources. Every letter will be answered. The authors will receive free copies of the journal. Papers in electronic form are accepted. They can be e-mailed in Microsoft Word 7.0a (or lower) for Windows 95, WordPerfect 6.0 (or lower) for Windows 95, or text files.

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C O N T E N T S ARTICLES: Micha Fleuren, Smarandache Factors and Reverse Factors ..... 5 Henry Ibstedt, Smarandache Continued Fractions ............. 39 Constantin Dumitrescu, Carmen Rocsoreanu, Some Connections Between the Smarandache Function and the Fibonnaci Sequence . 50 Y. V. Chebrakov, V. V. Shmagin, Computative Paradoxes in Modern Data Analysis ........................................ 61 Kevin Ford, The Normal Behavior of the Smarandache Function 81 J. Sándor, On Values of Arithmetical Functions at Factorials I ................................................ 87 Steven R. Finch, The Average Value of the Smarandache Function .................................................... 95 Sabin Tabirca, Smarandache Magic Squares ................... 100 Sabin Tabirc|, Tatiana Tabirc|, Some Inequalities Concerning Smarandache's Function ...................................... 105 Sebastián Martín Ruiz, Smarandache's Function Applied to Perfect Numbers ............................................. 114 Florian Luca, On the Divergence of the Smarandache Harmonic Series ...................................................... 116 Sabin Tabirc|, Tatiana Tabirc|, A Generalization of Euler's Function .................................................... 119 Sebastián Martín Ruiz, A Result Obtained Using Smarandache's Function .................................................... 123 J. Sándor, On An Inequality for the Smarandache Function...125 Florian Luca, On A Series Involving S(1)@S(2)...@S(n) ..... 128 Sebastián Martín Ruiz, A Congruence with Smarandache's Function .................................................... 130 Henry Ibstedt, An Integer as a Sum of Consecutive Integers . 133 NOTES: Mihaly Bencze, A New Inequality for the Smarandache Function Maohua Le, A Formula for the Smarandache Function .......... Maohua Le, On the Diophantine Equation S(n) = n ............ Maohua Le, On Smarandache Divisor Product .................. Maohua Le, On the Smarandache n-ary Sieve .................. Maohua Le, Perfect Powers in the Smarandache Permutation Sequence .................................................... Maohua Le, On Smarandache Pseudo-Powwers of Third Kind ..... Maohua Le, An Improvement on the Smarandache Divisibility Theorem ..................................................... Maohua Le, On Primes in the Smarandache Pierced Chain ...... 4

Maohua Le,

139 140 142 144 146 148 150 152 154

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Primes in the Sequences {nn+1}n=1 and {nn+1}n=1 .

156

Maohua Le, On the Smarandache Prime Additive Complement Sequence .................................................... 158 210

Mihaly Bencze, An Inequality for the Smarandache Function .. 160 Maohua Le, On Smarandache Simple Functions ................. 161 Charles Ashbacher, Maohua Le, On Smarandache Simple Continued Fractions ................................................... 163 Maohua Le, Notes on Primes Smarandache Progressions ........ 165 Maohua Le, The Primes p with lg(p) = 1 ..................... 167 Maohua Le, Some Solutions of the Smarandache Prime Equation 169 Charles Ashbacher, Smarandache Series Converges ............. 171 4

Maohua Le, Maohua Le, Maohua Le,

A Note on Primes in the Sequence {an+b}n=1 ...... 172 The Primes in the Smarandache Symmetric Sequence 174 On Smarandache General Continued Fractions ...... 176

Maohua Le, The Lower Bound for the Smarandache Counter C(0,n!) ..................................................... 178 Maohua Le,

On Smarandache Pseudo-Primes of Second Kind ....

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MATHEMATICAL HISTORY: Henry Ibstedt, Summary of the First International Conference on Smarandache Type Notions in Number Theory (University of Craiova, August 21-24, 1997) ................................ 181 Henry Ibstedt, Preambul to the First International Conference on Smarandache Type Notions in Number Theory (University of Craiova, August 21-24, 1997) ................................ 182 Henry Ibstedt, The Smarandache Sequence Inventory .......... 183 Henry Ibstedt, List of Smarandache Documents ............... 191 I. B|l|cenoiu, V. Seleacu, History of the Smarandache Function .................................................... 192 J. Castillo, Other Smarandache Type Functions .............. 202 M. L. Perez, Survey on the Research of Smarandache Notions 205 REVIEWS: Book Reviews ................................................ 207

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