Generalized Partitions and New Ideas On Number Theory and Smarandache Sequences Editor’s Note This book arose out of a collection of papers written by Amarnath Murthy. The papers deal with mathematical ideas derived from the work of Florentin Smarandache, a man who seems to have no end of ideas. Most of the papers were published in Smarandache Notions Journal and there was a great deal of overlap. My intent in transforming the papers into a coherent book was to remove the duplications, organize the material based on topic and clean up some of the most obvious errors. However, I made no attempt to verify every statement, so the mathematical work is almost exclusively that of Murthy. I would also like to thank Tyler Brogla, who created the image that appears on the front cover. Charles Ashbacher Smarandache Repeatable Reciprocal Partition of Unity {2, 3, 10, 15} 1/2 + 1/3 + 1/10 + 1/15 = 1 Amarnath Murthy / Charles Ashbacher AMARNATH MURTHY S.E.(E&T) WELL LOGGING SERVICES OIL AND NATURAL GAS CORPORATION LTD CHANDKHEDA AHMEDABAD GUJARAT- 380005 INDIA CHARLES ASHBACHER MOUNT MERCY COLLEGE 1330 ELMHURST DRIVE NE CEDAR RAPIDS, IOWA 42402 USA GENERALIZED PARTITIONS AND SOME NEW IDEAS ON NUMBER THEORY AND SMARANDACHE SEQUENCES Hexis Phoenix 2005 1 This book can be ordered in a paper bound reprint from: Books on Demand ProQuest Information & Learning (University of Microfilm International) 300 N. Zeeb Road P.O. Box 1346, Ann Arbor MI 48106-1346, USA Tel.: 1-800-521-0600 (Customer Service) http://wwwlib.umi.com/bod/search/basic Peer Reviewers: 1) Eng. Marian Popescu, Str. Spaniei, Bl. O8, Sc. 1, Ap. 14, Craiova, Jud. Dolj, Romania. 2) Dr. Sukanto Bhattacharya, Department of Business Administration, Alaska Pacific University, U.S.A. 3) Dr. M. Khoshnevisan, School of Accounting and Finance, Griffith University, Gold Coast, Queensland 9726, Australia. Copyright 2005 by Hexis., Amarnath Murthy and Charles Ashbacher Many books can be downloaded from the following E-Library of Science: http://www.gallup.unm.edu/~smarandache/eBooks-otherformats.htm Front cover image © Tyler Brogla and Charles Ashbacher ISBN: 1-931233-34-9 Standard Address Number: 297-5092 Printed in the United States of America 2 Editor’s Note This book arose out of a collection of papers written by Amarnath Murthy. The papers deal with mathematical ideas derived from the work of Florentin Smarandache, a man who seems to have no end of ideas. Most of the papers were published in Smarandache Notions Journal and there was a great deal of overlap. My intent in transforming the papers into a coherent book was to remove the duplications, organize the material based on topic and clean up some of the most obvious errors. However, I made no attempt to verify every statement, so the mathematical work is almost exclusively that of Murthy. I would also like to thank Tyler Brogla, who created the image that appears on the front cover. Charles Ashbacher 3 Table of Contents Editor’s Note ………………….. …………………… 3 Table of Contents …………………. …………………… 4 Chapter 1 Smarandache ………………. 11 Partition Functions 1.1 Smarandache ………………. 11 Partition Sets, Sequences and Functions 1.2 A Program to ………………. 16 Determine the Number of Smarandache Distinct Reciprocal Partitions of Unity of a Given Length 1.3 A Note On Maohua ………………. 18 Le’s Proof of Murthy’s Conjecture On Reciprocal Partition Theory 1.4 Generalization of ………………. 19 Partition Functions, Introduction of the Smarandache Partition Function 1.5 Open Problems and ………………. 28 Conjectures On the Factor/Reciprocal Partition Theory 1.6 A General Result on ………………. 30 the Smarandache Star Function 1.7 More Results and ………………. 40 Applications of the Generalized Smarandache Star Function 1.8 Properties of the ……………….. 47 Smarandache Star Triangle 1.9 Smarandache Factor ……………….. 50 4 Partitions Of A Typical Canonical Form 1.10 Length/Extent of ……………….. 55 Smarandache Factor Partitions 1.11 More Ideas On ………………… 58 Smarandache Factor Partitions 1.12 A Note On the ………………… 61 Smarandache Divisor Sequences 1.13 An Algorithm for ………………… 62 Listing the Smarandache Factor Partitions 1.14 A Program For ………………… 64 Determining the Number of SFPs 1.15 Chapter References ………………… 70 Chapter 2 Smarandache ………………… 72 Sequences 2.1 On the Largest Balu ………………… 72 Number and Some SFP Equations 2.2 Smarandache Pascal ………………… 75 Derived Sequences 2.3 Depascalization of ………………… 81 Smarandache Pascal Derived Sequences and Backward Extended Fibonacci Sequence 2.4 Proof the the ………………… 83 Depascalization Theorem 2.5 Smarandache ………………… 86 Friendly Numbers and A Few More Sequences 2.6 Some New ………………… 89 Smarandache Sequences, Functions and Partitions 5 2.7 Smarandache ………………… 94 Reverse Auto Correlated Sequences and Some Fibonacci Derived Smarandache Sequences 2.8 Smarandache Star ………………… 98 (Stirling) Derived Sequences 2.9 Smarandache ………………… 100 Strictly Staircase Sequence 2.10 The Sum of the ………………… 101 Reciprocals of the Smarandache Multiplicative Sequence 2.11 Decomposition of ………………… 103 the Divisors of A Natural Number Into Pairwise Co- Prime Sets 2.12 On the Divisors of ………………… 106 the Smarandache Unary Sequence 2.13 Smarandache Dual ………………… 108 Symmetric Functions and Corresponding Numbers of the Type of Stirling Numbers of the First Kind 2.14 On the Infinitude of ………………… 110 the Smarandache Additive Square Sequence 2.15 On the Infinitude of ………………… 111 the Smarandache Multiplicative Square Sequence 2.16 Another ………………… 113 Classification of the Ocean of 6 Smarandache Sequences 2.17 Pouring a Few ………………… 114 Drops in the Ocean of Smarandache Sequences and Series 2.18 Smarandache …………………. 118 Pythagoras Additive Square Sequence 2.19 The Number of …………………. 120 Elements the Smarandache Multiplicative Square Sequence and the Smarandache Additive Square Sequence Have in Common 2.20 Smarandache ………………… 121 Patterned Perfect Cube Sequences 2.21 The Smarandache ………………… 122 Additive Cube Sequence is Infinite 2.22 More Examples and ………………… 123 Results On the Infinitude of Certain Smarandache Sequences 2.23 Smarandache ………………… 124 Symmetric (Palindromic) Perfect Power Sequences 2.24 Some Propositions ………………… 126 On the Smarandache n2n Sequence 2.25 The Smarandache ………………… 128 Fermat Additive Cube Sequence 2.26 The Smarandache ………………… 130 nn2 Sequence Contains No Perfect 7 Squares 2.27 Primes in the ………………… 132 Smarandache nnm Sequence 2.28 Some Ideas On the ………………….. 133 Smarandache nkn Sequence 2.29 Some Notions On ………………….. 134 Least Common Multiples 2.30 An Application of ………………….. 137 the Smarandache LCM Sequence and the Largest Number Divisible by All the Integers Not Exceeding Its rth Root 2.31 The Number of ………………….. 138 Primes In the Smarandache Multiple Sequence 2.32 More On the ………………….. 139 Smarandache Square and Higher Power Bases 2.33 Smarandache Fourth ………………….. 140 and Higher Patterned/Additive Perfect Power Sequences 2.34 The Smarandache ………………….. 142 Multiplicative Cubic Sequence and More Ideas on Digit Sums 2.35 Smarandache Prime ………………….. 143 Generator Sequence 2.36 Chapter References 145 Chapter 3 Miscellaneous …………………. 147 Topics 3.1 Exploring Some …………………. 147 New Ideas On Smarandache Type Sets, Functions and Sequences 8 3.2 Fabricating Perfect ………………….. 155 Squares With a Given Valid Digit Sum 3.3 Fabricating Perfect …………………… 157 Cubes With a Given Valid Digit Sum 3.4 Smarandache …………………… 161 Perfect Powers With Given Valid Digit Sum 3.5 Numbers That Are a …………………… 164 Multiple of the Product of Their Digits and Related Ideas 3.6 The Largest and ………………….. 166 Smallest mth Power Whose Digit Sum/Product Is It’s mth Root 3.7 A Conjecture on …………………. 168 d(N), the Divisor Function Itself As A Divisor With Required Justification 3.8 Smarandache …………………. 171 Fitorial and Supplementary Fitorial Functions 3.9 Some More …………………. 174 Conjectures On Primes and Divisors 3.10 Smarandache 176 Reciprocal Function and An Elementary Inequality 3.11 Smarandache …………………. 178 Maximum Reciprocal Representation Function 3.12 Smarandache …………………. 179 Determinant Sequence 9 3.13 Expansion of xn in …………………. 182 Smarandache Terms of Permutations 3.14 Miscellaneous …………………. 187 Results and Theorems on Smarandache Terms and Factor Partitions 3.15 Smarandache- ………………… 192 Murthy’s Figures of Periodic Symmetry of Rotation Specific to an Angle 3.16 Smarandache Route …………………. 198 Sequences 3.17 Smarandache …………………. 200 Geometrical Partitions and Sequences 3.18 Smarandache Lucky …………………. 207 Methods in Algebra, Trigonometry and Calculus 3.19 Chapter References …………………. 210 Index 212 10 Chapter 1 Smarandache Partition Functions Section 1 Smarandache Partition Sets, Sequences and Functions Unit fractions are fractions where the numerator is 1 and the denominator is a natural number. Our first point of interest is in determining all sets of unit fractions of a certain size where the sum of the elements in the set is 1. Definition: For n > 0, the Smarandache Repeatable Reciprocal partition of unity for n (SRRPS(n)) is the set of all sets of n natural numbers such that the sum of the reciprocals is 1. More formally, n SRRPS(n) = { x | x =(a1, a2, . ,an) where Σ 1/ar = 1 }. r=1 fRP(n) = order of the set SRRPS(n). For example, SRRPS(1) = { (1) }, fRP(1) = 1. SRRPS(2) = { (2,2)}, fRP(2) = 1. (1/2 + 1/2 = 1). SRRPS(3) = { (3,3,3), (2,3,6), (2,4,4)}, fRP(3) = 3. SRRPS(4) = { (4,4,4,4), (2,4,6,12),(2,3,7,42),(2,4,5,20),(2,6,6,6),(2,4,8,8), (2,3,12,12),(4,4,3,6),(3,3,6,6),(2,3,10,15),(2,3,9,18)}, fRP(4) = 14. Definition: The Smarandache Repeatable Reciprocal Partition of Unity Sequence is the sequence of numbers SRRPS(1), SRRPS(2), SRRPS(3), SRRPS(4), SRRPS(5), . Definition: For n > 0, the Smarandache Distinct Reciprocal Partition of Unity Set (SDRPS(n)) is SRRPS(n) where the elements of each set of size n must be unique. More formally, n SRRPS(n) = { x | x =(a1, a2, . ,an) where Σ 1/ar = 1 and ai = aj <=> i = j }.