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...F. Smarandache has defined many sequences of integers formed by concatenating the positive integers in specific ways. A question that is often asked about most of these sequences concerns the number of primes in the sequence. As Paul Erdos pointed out, d...

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...Using the bases from three through nine to form new sequences, we can then ask the same question regarding the number of primes in the sequence. That appears to be a question just as difficult as the one for base ten numbers, and is summarized in the following statement....

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...Preface.3 -- Table of Contents.4 -- Chapter 1 Sequences Formed by Concatenating All the Positive Integers.5 -- Chapter 2 Sequences Formed by Concatenating Selected Natural Numbers .44 -- Chapter 3 Smarandache Stereograms .70 3.1 -- The Stereogram.70 -- 3.2 Altering the B...

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...CHARLES ASHBACHER Smarandache Sequences, Stereograms and Series 1111111111111111111111111111111111111... ...Hexis Phoenix 2005 Charles Ashbacher Mount Mercy College Smarandache Sequences, Stereograms and Series Hexis Phoenix 2005 2 This book can be or... ... Preface ………………. …………………. 3 Table of Contents ………………. …………………. 4 Chapter 1 Sequences Formed by Concatenating All the Positive Integers …………………. 5 Chap... ...ces Formed by Concatenating All the Positive Integers …………………. 5 Chapter 2 Sequences Formed by Concatenating Selected Natural Numbers …………………. 44 Chap... ...Smarandache Function …………………... 95 Index …………………. ………………… 133 5 Chapter 1 Sequences Formed By Concatenating All the Positive Integers F. Smarandache... ...through n, so the probability that they are prime is higher than that of a randomly selected odd integer. The elements of the Permutation Sequence ar... ...ncatenation of the primes modulo numbers, prime or not prime, appear to be randomly distributed. Once the issue of the apparent appearance of random ... ...ent with the idea that the elements of the Concatenated Prime Sequence are randomly distributed around the prime numbers. 4) Smarandache Concatenated ...

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..., paper everywhere, the innards of the computer and its disemboweled printer strewn randomly across the carpet. Even the old men’s suits were not sp... ...nd put some thinking distance between me and this bizarre affair. I assembled a random sample of attire, stuffed it into a battered suitcase, and... ... the same, simple question. They attacked every angle, made me go through the whole sequence of events time and again. But they always came back to ... ...orite couch and placed one bare and hairy foot over another. “Digital images are sequences of bits. Some sequences designate colors, other sequen... ... Let’s assume that you isolate the last 2 bits in every color component, in all the sequences that code for colors. By carefully replacing these las... ...ut otherwise nothing seemed to have been disturbed. My heart raced as I picked up a random assortment of bath towels, T-shirts, underwear, jackets, ... ...where he was going. “What if the stegotexts, these senseless rivers of apparently random characters, were actually somehow linked to one another? ... ...e, or set of rules with many steps. When you follow the algorithm you generate two sequences, two cryptographic keys. It’s like having a single key ... ...rime numbers used to generate the public key are large enough, the result is a long sequence of figures. You can try to find out the factors – the p...

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...ther necessitates the effect, or is a sufficient condition for its occurrence. The sequence is either inevitable, or possible. The meaninglessness ... ...tations, including those related to human life directly. We even issue patents for sequences of human DNA. And people do own forests and rivers and... ... beings. On the contrary, complexity usually points towards a natural source and a random origin. It is also true that complexity and artificiality...

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...n 1956 IBM invented the first computer disk storage system, the 305 RAMAC (Random Access Method of Accounting and Control). This system could store ... ... the process that moves the read/write heads to the correct cylinder on a random access on the hard disk. Therefore, at the time that the heads arri... ... that represents the composite of all the other specifications reflecting random performance positioning in the hard disk. As such, it is the best fi... ...lso depends on the access pattern of the particular program being run (the sequence of addresses being read and written). Caches rely on two properti...

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...n 1956 IBM invented the first computer disk storage system, the 305 RAMAC (Random Access Method of Accounting and Control). This system could store ... ... the process that moves the read/write heads to the correct cylinder on a random access on the hard disk. Therefore, at the time that the heads arri... ... that represents the composite of all the other specifications reflecting random performance positioning in the hard disk. As such, it is the best fi... ...lso depends on the access pattern of the particular program being run (the sequence of addresses being read and written). Caches rely on two properti...

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...ies: an even-number (α-particle) progression in nucleo protons and Fibonacci sequence in the outer proton, (octet and face) sites. The projection of ... ...of light in refractive transmission is attributed to an absorption-emission sequence of the light by atoms of the matter field; in which, successive... ...atter), (diffuse reflection), is defined as incident VLS being reflected at random angles due to uneven contours, (height: h 1 - h 2 > λ), in the m... ... in the matter field surface. The typically non-opalescent effect indicates randomness of angle reflection ≅ 100%. This indicates two modes of refle... ...0%. This indicates two modes of reflection having very different degrees of randomness in reflection angle affect. Specular: the light is reflected ... ...ble) scale. SQP-1 Conclusions in MT To summarize, the absorption-emissive sequence in refraction imparts no visible atomic field to the radiation,... ...ide molecular structural information from diffractive effects, but the high randomness in diffuse reflection angles precludes this possibility. Lumin...

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...rom Egypt was made prematurely, and that there are farfetched historical con- sequences to it. Did the premature liberation prevent a possibility to ... ...over the world? Because drugs create an illusion of freedom, with severe con- sequences. Man feels that he is in a completely different situation; he... ...“molds.” A hologram may sometimes serve as a good example to stress that each randomly taken part of the spiritual realm consists of all the parts, ... ...of time and where does it come from? A: There is a spiritual time, which is a sequence of situations (you can read about it in the first part of The... ...her name from there. That is quite sufficient. I don’t advise giving children random names designating objects of nature and the like as is fashiona...

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...dividuals finding themselves separately cast onto lifeboats in an ocean of random waves for they found oceans of thoughts within themselves that seeme... ...ceived them as perceiving him to be. There were also briefer skits in the random feelings, thoughts, and perceptions he was trying to categorize. On... ...rother. He felt the insect monster inject him with the malaria of tragedy: random images were kicked about in his mind like starving dogs allowed to p... ...ignorant to not know that or male. --He chitchatted, my dear, in a logical sequence that was a bit desultory at times. Humans call such an inexact ord... ...imself in all of the spinning images: sounds, smells, and visions all spun randomly. Finally there was a bit of a shape and texture to his thinking an... ... Perhaps they needed to believe in the continuum of action and being (the random balls of matter that they were). Inside the stadium-shaped park wer...

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...e that kind of a thing. It is the other way around. It hits but it hits at random. That is the way nature operates, lightning hits you somewhere. It ... ...its and is helping the people by providing shade, etc. It just strikes at random. In exactly the same way, it happens to a particular individual. ... ...If this is so then where is the room for practices and volition? It comes randomly, not because you deserve it. If you are lucky enough to have th... ...single event. It is the logical, cause and effect thinking that imposes a sequence to events. The sudden evidence of light and the throwing of the l...

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...sionmark 200 2nd Ave. SE Cedar Rapids, IA 52401 Chapter 1 Smarandache Sequences In the time since it was first published[ 1], the Smarandache fu... ...order. In this first chapter, we will concentrate on notions that involve sequences, moving on to other, more specific points in the second. The fi... ...t problem we will deal with is number (6) in [5]. Smarandache permutation sequence: 12,1342.135642, 13578642, 1357910864~ 135791112108642,. Or as a... ...)(2n)(2n-2)(2n-4) 642 where SPS is an acronym for Smarandache Permutation Sequence. Note on notation: As a general principle, acronyms will be creat... ...ect of the next theorem. Theorem 10: If a positive integer n is chosen at random, the odds are overwhelming that n is a pseudo-odd number of the fir... ... 4 * 5'-[ k-digit numbers containing no odd number. The probability of a random k-digit number containing all even numbers is then 4*5"-: _ .f --... ... the digits is odd. Theorem 11: If a positive even integer n is chosen at random, the odds are overwhelming that n is a pseudo-odd number of the sec... ...e digits is an odd number. It should be clear that the probability that a randomly chosen positive integer is pseudo- odd of the third kind is also o... ...nition is the following. Theorem 12: If a positive integer n is chosen at random, the probability that n is both a pseudo-odd number of the first ki...

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...ing. 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. Moreover a chapter (IV) is dedicated to the analysis of Smarandache Double factorial function introduced in number theory by F. Smarandache ("The Florentin Smarandache p...

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...A SET OF NEW SMARANDACHE FUNCTIONS, SEQUENCES AND CONJECTURES IN NUMBER THEORY Felice Russo AMERICAN RESEARCH P... ...functions, new conjectures, solved/unsolved problems, new Smarandache type sequences and new Smarandache Notions in number theory. Moreover a chapter ... ...respectively) values. For the first 1000 values of Zt(n) the largest found sequences have k=5 and k=4 respectively. Example: Z... ...repeated application of the Zt function. An n-cycle, instead, is a finite sequence of points 1 , 1 0 , - n Y Y Y K such that, under the function Zt,... ...Totient numbers. Problem 37. Is the number 0.1243549107585.... where the sequence of digits is Zt(n) for 1 ‡ n an irrational or trascendental number... ...77 The Prime Number Theorem [4] states (roughly) that the probability of a randomly chosen number of the size of n being prime is about: ... ...act this estimate has to be increased since such numbers are not chosen at random; they are specifically chosen not to be divisible by 2 or 5 so the ...

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...nt of power p in n. That is, ep(n) = m, if pm j n and pm+1 - n. In problem 68 of [1], Professor F.Smarandache asked us to study the properties of the sequence fep(n)g. About the elementary properties of this function, many scholars have studied it (see reference [2]-[7]), and got some useful results. For examples, Liu Yanni [2] studied the mean value properties of ep(bk(n)...

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...ive function 18 X. Pan and P. Zhang : An identity involving the Smarandache function ep(n) 26 A.A.K. Majumdar : A note on the Smarandache inversion sequence 30 F. Li : Some Dirichlet series involving special sequences 36 Y. Wang : An asymptotic formula of Sk(n!) 40 S. Gao and Z. Shao : A fuzzy relaxed approach for multi-objective transportation problem 44 J. Li : An ...

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...tems, multi-metric spaces,. . . , etc.. Smarandache geometries; Differential Geometry; Geometry on manifolds; Topological graphs; Algebraic graphs; Random graphs; Combinatorial maps; Graph and map enumeration; Combinatorial designs; Combinatorial enumeration; Low Dimensional Topology; Differential Topology; Topology of Manifolds; Geometrical aspects of Mathematical Phy...

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...Sequences on Graphs with Symmetries Linfan Mao Chinese Academy of Mathematics and System Science, Beijing, 10080, P.R.China Beijing Institute of Civil Engineering and Architecture, Beijing, 100044, P.R.China E-mail: maoli...

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...Contents Lucas Graceful Labeling for Some Graphs BY M.A.PERUMAL, S.NAVANEETHAKRISHNAN AND A.NAGARAJAN . . . . . 01 Sequences on Graphs with Symmetries BY LINFAN MAO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Supermagic Coverings of Some Simple Graphs BY P.JEYANTHI AND P.SELVAGOPAL . . . . . . . . . . . . . ....

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... it is being tragically eroded by our current copyright, patent, and trademark laws. In a series of fascinating case studies, Boyle explains why gene sequences, basic business ideas and pairs of musical notes are now owned, why jazz might be illegal if it were invented today, why most of 20th century culture is legally unavailable to us, and why today's policies would prob...

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... should be the farthest from the market. In stories about stem cell and gene sequence patents, critics have mused darkly about the way in which the st... ...y interfere with your plans to use it for growing crops. By contrast, a gene sequence, an MP3 file, or an image may be used by multiple parties; my The... ...t 49 ___-1 ___0 ___ 1 37278_u01.qxd 8/28/08 11:04 AM Page 49 layer; gene sequence patents come very close to being rights over a particular discov... ...hich simply means that you strike the notes of the chord individually and in sequence. It was, in fact, a pretty standard “deedly” sound, familiar fro... ... The original beats have a totemic significance—like the great standard chord sequences in jazz. One cannot substitute replicas for James Brown’s funki... ...hat these questions are fun to debate but ultimately irrelevant. 12 Assume a random distribution of incentive struc- tures in different people, a glob... ...effectively scattered U.S. approach to data privacy, for example, pro- duces random islands of privacy protection in a sea of potential vulnerability.... ...e Definitive History of the Most Important Artists and Their Music (New York: Random House, 1992), 130. Notes to pages 127–130 281 ___-1 ___0 ___ 1 372... ...e Future of Ideas: The Fate of the Com- mons in a Connected World (New York: Random House, 2001), and Lawrence Lessig, Free Culture (New York: Penguin...

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...re of principal lters on po-semigroups 50 Y. Lu : F. Smarandache additive k-th power complements 55 M. Le : The Smarandache reverse auto correlated sequences of natural numbers 58 M. Karama : Smarandache partitions 60 L. Mao : On Algebraic Multi-Group Spaces 64 F. Russo : The Smarandache P and S persistence of a prime 71 Y. Lu : On the solutions of an equation involv...

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...ays, takes us into seeming irrelevance - maybe as a means of escapism, and all by unpunctuated expertly crafted emotionally evocative (and) beautiful sequences of words. It's akin to poetry, or song lyrics at times. Words are used rather like musical notes, arranged in a sensitive, sometimes aggressive ways, constructing and deconstructing, effectively facilitating your ow...

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... a way to say how it is to continue to go on there is a pattern to deterioration as there is to horror and escape as there is to relating a series of random occurrences that are bound together by the problem of consciousness there is a way to wait in a hospital through moments descriptions of infirmities and remedies that do not end but go on there are the reasons there is...

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...al wealth at time t = T, Z 0 is the initial wealth at time t = 0, x is a random loss variable, I(x) is the indemnification function, C(x) is the co... ...s that the generalized financial market cannot possibly be reduced to pure random walk if we do away with the assumption of normality, it does not ne... ...w information. Some inefficiency however may also be attributed to purely random white noise unrelated to any coherent market information. If the pr... ...al price-path meaning that all market information (as well as some of the random white noise) could become automatically anchored to this perceived ... ...lied volatility on short- term options with probability 0.99 or out of (C) random white noise with probability of 31 0.01. The other sees it as ei... ...etter j will be received. A code word of length n is defined as a sequence of n input letters, which are actually n integers chosen from 1, ... ...= 1, 2 … n. As per Shannon’s coding theorem, it is possible to encode all sequences of n message integers into sequences of binary digits in such a w... ....A., 1984. Gerard, H. B. Choice difficulty, dissonance and the decision sequence. Journal of Personality 35, 1967: 91-108. Goldberg, D. E. Gen...

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...Probability (called sometimes the objective probability) process uncertainty of random type (stochastic) introduced by the chance. Uncertainty of th... ...t result. This might be comparable to approximating the limit of a convergent sequence, calculating more and more terms, or by calculating the limit... ...be a finite discrete set of exhaustive and exclusive hypotheses or outcomes of a random experiment. The probability of has been defined and interpreted ... ...just know that the cardinality numbers of follow the integers of the Dedekind’s sequence minus one when Card increases. Examples 1. If we consider (em... ...ations in the mathematical community. Cardinality numbers follow the Dedekind’s sequence (minus one), i.e. when Card . 4.3 General basic assignment n... ...onmental Studies, Duke University, Durham, NC, 1986. [35] Li X.R.,“Probability, Random Signals, and Statistics”,CRC Press, 1999. [36] Mahler R., “Comb... ...Functions”, Comm. Pure Appl. Math., 23, pp. 299 312, 1970. [55] Sloane N.J.A., “Sequences A006826/M2469, A007153/M3551, and A014466”, in “An On Line V... ...imum Deviation=3% Average Deviation=.1 Neural network learning method: Batch Random Step width =.03 Winner Neuron=4 Winner Neuron=4 Winner N... ...best compromise is used in most control applications. The rule block is 1 with random Dos value generator. 5.Conclusions: 5.Conclusions: 5.Conclus...

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...nized and increasing structural and functional adaptability has been attributed to random-stochastic processes (such as environmentally-induced gen... ...eter) against Disorder. In this context, natural selection is no more "blind" or "random" than its subjects. It is discriminating, encourages stru... ...hance happening. We are the culmination of orderly processes - not the outcome of random events. The Universe enables us and our world because - a... ...nt. Therefore, we are forced to conclude that quantum events are guided by some non-random principle (such as the increase in order). This, exactly,... ... beings. On the contrary, complexity usually points towards a natural source and a random origin. Complexity and artificiality are often incompatib... ...re carefully "selected" for some purpose, within a certain pattern and in a certain sequence. What could that purpose be? Probably, the extension an... ...ecause a minimum number of axioms and hypotheses yield, in an inexorable deductive sequence, everything else known in the field the theory pertains... ..."meaning". Both Donald Davidson and Alfred Tarski suggested that truth exists where sequences of objects satisfy parts of sentences. The meanings of... ...ecause a minimum number of axioms and hypotheses yield, in an inexorable deductive sequence, everything else known in the field the theory pertains...

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... working years at the Gazette and in retirement, curiosity prompted notes randomly taken and linked to history trails recalled from high school and ... ... Europe. Expanded Access To Wasted Intellect Feeding my curiosity with random research convinced me of the following: • Earlier Inf... ...eyond geometry— fixes the zero as the starting linchpin in plus and minus sequences. History of the missing number hazy Mathematicians find the ... ...rint shop helpers were still known before the advent of union apprentices—randomly and appropriately tossed it into a ―hell box.‖ Later, the prin... ...third digits and then by a decimal point and more numbers. (Note that the sequence recognizes the proper placement of zero by starting with 000 rath... ...or ETAOIN SHRDLU to be listed in the Oxford English Dictionary and in the Random House Webster's Unabridged Dictionary. CHAPTER 21 ―Einstein wasn’... ...imon and Schuster, 2001. Boorstin, Daniel J. The Discoverers. New York: Random House, 1983. Broder, David. Behind the Front Page: A Candid Look ...