Smarandache Notions Journal, Vol. 12

Vol. 12, No. 1-2-3, Spring 2001

ISSN 1084-2810

SMARANDACHE NOTIONS JOURNAL

TRANSILVANIA UNIVERSITY OF BRAÔOV

The scheme on the first cover represents the Smarandache function 7-7 additive relationships and belongs to H. Ibstedt (see p. 73). This journal is an extension of the "Smarandache Function Journal", ISBN 1053-4792, Vol. 1-6, 1990-1995. SNJ is yearly published (in the Spring or Fall) in a 300-400 pages volume, and 800-1000 copies. SNJ 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), "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: Murad A. AlDamen, P. O. Box 713330, Hai Nazzal 11171, Amman, Jordan. Y. V. Chebrakov, Department of Mathematics, St.-Petersburg Technical University, Nevsky 3-11, 191186, Russia. Ion Goian, Department of Algebra, Number Theory and Logic, State University of Kishinev, Republic of Moldova. Florian Luca, Instituto de Matematicas UNAM, Campus Morelia, Apartado Postal 61-3 (Xangari), CP 58 089, Morelia, Michoacan, México. Chris Nash, 5000 Bryan Station Road #5, Lexington, KY 40516-9505, USA. Carlos B. Rivera F., J. Pardavé 517, Col Lomas del Roble, San Nicolás de los Garza, CP 66450, Nuevo León, México. Sebastian Martin Ruiz, Avda. De Regla, 43, Chipiona 11550, Cadiz, Spain. Copyright 2001 by Transilvania University of BraÕov, Department of Mathematics and Computer Science, BraÕov, Romania. ASSOCIATE EDITORIAL BOARD: Charles Ashbacher, Charles Ashbacher Technologies, Box 294, 119 Northwood Drive, Hiawatha, Iowa 52233, USA. Ion Balacenoiu, Department of Mathematics, University of Craiova, Romania. Mihaly Bencze, Str. H|rmanului 6, 2212 S|cele 3, Jud. BraÕov, 2

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. Leonardo Fernandes Dias da Motta, Av. Conselheiro Furtado, 1574, Apt. 501, Bairro Batista Campos, Belem, PA 66040-100, Brazil. Mike Mudge, 22 Gors Fach, Pwll-Trap, St Clears, Carmarthenshire, Dyfed SA33 4AQ, United Kingdom. Jean Normand, 3, square Robinson, 92330 Sceaux, France. E. R|descu & N. R|descu, Department of Mathematics, University of Craiova, Romania. Francisco Bellot Rosado, Dos de Mayo 16-8E dcha, E-47004 Valladolid, Spain. Felice Russo, Micron Technology Italy, Avezzano (Aq), Italy. Jozsef Sandor, Department of Mathematics, BabeÕ-Bolyai University, 3400 Cluj-Napoca, Romania. Vasile Seleacu, Department of Mathematics, University of Craiova, 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. INDICATION TO AUTHORS Authors of papers concerning any of Smarandache type functions, numbers, sequences, integer algorithms, paradoxes, Non-Euclidean geometries, etc. are encouraged to submit manuscripts to the Editors: Prof. Sabin & Tatiana Tabirca Department of Mathematics Transilvania University of Brasov, Romania. The submitted manuscripts may be in the format of remarks, 3

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 writing 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 97 (or lower), WordPerfect 7.0 (or lower), PDF 4.0, or ASCII files.

4

C O N T E N T S

MATHEMATICS: ROBERTO TORRETTI, A Model for Smarandache's Anti-Geometry .... 5 FLORIAN LUCA, The Average Smarandache Function ............... 19 TATIANA TABIRCA, SABIN TABIRCA, A Parallel Loop Scheduling Algorithm Based on the Smarandache f-Inferior Part Function .. 28 HENRY IBSTEDT, On the Pseudo-Smarandache Function and Iteration Problems. Part I: The Euler φ Function ...................... 36 HENRY IBSTEDT, On the Pseudo-Smarandache Function and Iteration Problems. Part II: The Sum of Divisors Function ............. 44 SABIN TABIRCA, Erdös-Smarandache Moments Numbers ............. 49 SABIN TABIRCA, TATIANA TABIRCA, Erdös-Smarandache Numbers .... 54 J. SÁNDOR, On the Pseudo-Smarandache Function ................ 59 HENRY IBSTEDT, Smarandache k-k Additive Relationships ........ 62 KRASSIMIR T. ATANASSOV, Remarks on Some of the Smarandache's Problems. Part 1 ............................................ 82 HENRY IBSTEDT, A Brief Account on Smarandache 2-2 Substractive Relationships ................................................ 99 HENRY IBSTEDT, On a Smarandache Partial Perfect Additive Sequence .................................................... 103 NIKOLAI NIKOLOV, KRASSIMIR ATANASSOV, On the 107-th, 108-th and 109-th Smarandache's Problems ............................... 108 KRASSIMIR T. ATANASSOV, On the 20-th and the 21-st Smarandache's Problems .................................................... 111 SABIN TABIRCA, TATIANA TABIRCA, On the Primality of the Smarandache Symmetric Sequences ............................. 114 KRASSIMIR T. ATANASSOV, On Four Prime and Coprime Functions . 122 Y. V. CHEBRAKOV, V. V. SHMAGIN, Investigating Connections Between Some Smarandache Sequences, Prime Numbers and Magic Squares . 126 TIANG ZHENGPING, XU KANGHUA, On Smarandache Sequences and Subsequences ................................................ 146 FELICE RUSSO, On Three Problems Concerning the Smarandache LCM Sequence .................................................... 153 FELICE RUSSO, On a Problem Concerning the Smarandache Unary Sequence .................................................... 156 FELICE RUSSO, An Introduction to the Smarandache Double Factorial Function .................................................... 158 ADRIAN VASIU, ANGELA VASIU, Geometrie Interioar| ............ 168 FELICE RUSSO, On Some Smarandache Conjectures and Unsolved 363

Problems ................................................... 172 FELICE RUSSO, A Recurrence Formula for Prime Numbers Using the Smarandache or Totient Functions ............................ 193 FELICE RUSSO, On Two Problems Concerning Two Smarandache P-Partial Digital Sequences ................................. 198 MIHALY BENCZE, Open Questions for the Smarandache Function .. 201 MAOHUA LE, On Smarandache Algebraic Structures. I: The Commutative Multiplicative Semigroup A(a,n) ................. 204 MAOHUA LE, On Smarandache Algebraic Structures. II: The Smarandache Semigroup ....................................... 207 MAOHUA LE, On Smarandache Algebraic Structures. III: The Commutative Ring B(a,n) ..................................... 209 MAOHUA LE, On Smarandache Algebraic Structures. IV: The Commutative Ring C(a,n) ..................................... 211 MAOHUA LE, On Smarandache Algebraic Structures. V: Two Classes of Smarandache Rings ........................................ 213 MAOHUA LE, A Note on the Smarandache Bad Numbers ............ 215 MAOHUA LE, A Lower bound for S(2p-1(2p-1)) ................... 217 MAOHUA LE, The Squares in the Smarandache Higher Power Product Sequences ................................................... 21364 MAOHUA LE, The Powers in the Smarandache Square Product Sequences ............................................................. 221 MAOHUA LE, The Powers in the Smarandache Cubic Product Sequences ............................................................. 223 MAOHUA LE, On the Smarandache Uniform Sequences ............. 226 MAOHUA LE, The Primes in the Smarandache Power Product Sequences of the Second Kind .......................................... 228 MAOHUA LE, The Primes in the Smarandache Power Product Sequences of the First Kind ........................................... 230 k MAOHUA LE, On the Equation S(mn)=m S(n) .................... 232 MAOHUA LE, On an Inequality Concerning the Smarandache Function ............................................................. 234 MAOHUA LE, The Squares in the Smarandache Factorial Product Sequence of the Second Kind ................................ 236 MAOHUA LE, On the Third Smarandache Conjecture about Primes . 238 MAOHUA LE, On Russo's Conjecture about Primes ............... 240 MAOHUA LE, A Conjecture Concerning the Reciprocal Partition Theory ...................................................... 242 MAOHUA LE, A Sum Concerning Sequences ....................... 244 J. SÀNDOR, A Note on S(n2) .................................. 246 J. SÀNDOR, On A New Smarandache Type Function .............. 247 MIHALY BENCZE, About the S(n)=S(n-S(n)) Equation ........... 249 SAM ALEXANDER, A Note on Smarandache Reverse Sequence ...... 250 364

AMARNATH MURTHY, Smarandache Pascal Derived Sequences ...... 251 AMARNATH MURTHY, Depascalisation of Smarandache Pascal Derived Sequences and Backward Extended Fibonacci Sequence ......... 255 AMARNATH MURTHY, Proof of the Depascalisation Theorem ...... 258 JÓZSEF SÁNDOR, On Certain Arithmetic Functions ............. 260 AMARNATH MURTHY, Smarandache Star (Stirling) Derived Sequences ............................................................. 262 AMARNATH MURTHY, Smarandache Friendly Numbers and A Few More Sequences .................................................. 264 AMARNATH MURTHY, Smarandache Geometrical Partitions and Sequences ............................................................. 268 AMARNATH MURTHY, Smarandache Route Sequences ............... 272 AMARNATH MURTHY, Smarandache Determinant Sequences ......... 275 AMARNATH MURTHY, Smarandache Reverse Auto Correlated Sequences and Some Fibonacci Derived Smarandache Sequences ........... 279 AMARNATH MURTHY, Smarandache Strictly Stair Case Sequences . 283 CSABA BÍRÓ, About a New Smarandache-type Sequence .......... 284 LEONARDO MOTTA, On the Smarandache Paradox ................. 287 SEBASTIÁN MARTÍN RUIZ, New Prime Numbers ................... 289 TAEKYUN KIM, A Note on the Value of Zeta ................... 291 FELICE RUSSO, Ten Conjectures on Prime Numbers ............. 295 ADRIAN VASIU, ANGELA VASIU, On Some Implications of Formalized Theories in our Life ....................................... 297 AMARNATH MURTHY, Decomposition of the Divisors of A Natural Number Into Pairwise Co-Prime Sets ......................... 303 AMARNATH MURTHY, Some Notions on Least Common Multiples .... 307 AMARNATH MURTHY, Smarandache Dual Symmetric Functions and Corresponding Numbers of the Type of Stirling Numbers of the First Kind ................................................. 309 AMARNATH MURTHY, Some More Conjectures on Primes and Divisors 311 MAOHUA LE, The Reduced Smarandache Square-Digital Subsequence is Infinite ................................................... 313 MAOHUA LE, The Reduced Smarandache Cube-Partial-Digital Subsequence is Infinite .................................... 315 MAOHUA LE, The Convergence Value and the Simple Continued Fractions of Some Smarandache Sequences .................... 317 MAOHUA LE, The First Digit and the Trailing Digit of Elements of the Smarandache Deconstructive Sequence .................... 319 MAOHUA LE, The 2-Divisibility of Even Elements of the Smarandache Deconstructive Sequence .................................... 321 MAOHUA LE, Two Smarandache Series .......................... 323 MAOHUA LE, The 3-Divisibility of Elements of the Smarandache Deconstructive Sequence .................................... 325 MAOHUA LE, Two Conjectures Concerning Extents of Smarandache 365

Factor Partition ........................................... 328 MAOHUA LE, On the Balu Numbers ............................. 331 MAOHUA LE, The Limit of the Smarandache Divisor Sequences .. 335 KRASSIMIR ATANASSOV, HRISTO ALADJOV, A Generalized Net For Machine Learning of the Process of Mathematical Problems Solving. / On An Example with A Smarandache Problem ................. 338 G. L. Waghmare, S. V. More, New Smarandache Algebraic Structures ............................................................. 344 DWIRAJ TALUKDAR, Lattice of Smarandache Groupoid ........... 347

PHYSICS: LEONARDO F. D. DA MOTTA, Smarandache Hypothesis: Evidences, Implications and Applications .............................. 354 LEONARDO F. D. DA MOTTA, Smarandache-Rodrigues-Maiorino (SRM) Theory ..................................................... 359 GHEORGHE NICULESCU, Quantum Smarandache Paradoxes .......... 361

366

The "Smarandache Notions Journal" is online at:

http://www.gallup.unm.edu/~smarandache/ and selected authors are manuscripts, the Internet

papers are periodically added to our web site. The pleased to send, together with their hard copy a floppy disk with HTML or ASCHII files to be put in as well.

Papers in electronic form are accepted. They can be e-mailed in Microsoft Word 97 (or lower), WordPerfect 7.0 (or lower), PDF 4.0, or ASCII files. Starting with this Vol. 10, the "Smarandache Notions Journal" is also printed in hard cover for a special price of $79.95.

$69.90