ABSTRACTS AND CONJECTURES ON SMARANDACHE NOTIONS 1) Palindromic Numbers and Iterations of the Pseudo-Smarandache Function A number is called palindromic if it reads the same forwards and backwards. For examples: 121, 34566543. 1111. The Pseudo-Smarandache Function Z(n) is defined for any n >= 1 as the smallest integer m such that n evenly divides 1 + 2 + ... + m. There are some palindromic numbers n such that Z(n) is also palindromic: Z(909) = 404, Z(2222) = 1111. k 0 Let Z (n) = Z(Z(Z(...(n)))), where the function Z is executed k times. Z (n) is, by convention, n. k Unsolved Question: What is the largest value of m so that for some n, Z (n) is a palindrome for all k = 0, 1, 2, ..., m? k Conjecture: There is no largest value of m such that for some n, Z (n) is a palindrome for all k = 0, 1, 2, ..., m. (C. Ashbacher) Reference: [1] Kashihara, Kenichiro, "Comments and Topics on Smarandache Notions and Problems", Erhus Univ. Press, Vail, USA, 1996. 2) Computational Aspect of the Smarandache's Function This note presents an algorithm, which tries to avoid the factorials, for the Smarandache's function computation. The complexity of the algorithm is studied using the main properties of the function. An interesting inequality is found giving the complexity of the function on the set {1, 2, ..., n}. 3) Some Upper Bounds for the Smarandache Function Average _ Let S = (1/n) {S(1) + S(2) + ... + S(n)} be the Smarandache Function Average. We prove that _ S <= (3/8)n + 1/4 + 2/n, for n > 5; _ S <= (21/72)n + 1/12 - 2/n, for n > 23; and we conjecture that _ S <= (2n)/(ln n), for n > 1, that we have checked with a C program for all numbers <= 10000. (S. Tabirca, T. Tabirca) 4) Another Conjecture on Prime Numbers One proves that the Smarandache Reverse Sequence 1, 21, 321, 4321, 54321, 654321, 7654321, ... (obtained by concatenation of natural numbers in a decreasing order) doesn't contains infinitely prime terms. 5) A conjecture on Smarandache Anti-Symmetric Sequence In this paper we study a conjecture which states the Smarandache Anti-Symmetric Sequence: ______________ 123...n123...n, for n >= 1, has no perfect-power term. 6) Analytical Formulas of 6 Smarandache Series and Their Application in the Magic Square Theory We present a set of analytical formulas for the computation of the general term in each of the following sequences: 1, 12, 123, 1234, 12345, 123456, ... (smarandache consecutive sequence) 1, 11, 121, 1221. 12321, 123321, ... (smarandache symmetric sequence) 1, 212, 32123, 4321234, 543212345, ... (smarandache mirror sequence) 1, 23, 456, 7891, 23456, 789123, 4567891, ... (smarandache deconstructive sequence) 1, 12, 21, 123, 231, 312, 1234, 2341, 3412, 4123, ... (smarandache circular sequence) 12, 1342, 135642, 13578642, 13579108642, ... (smarandache permutation sequence) in order to construct 3x3 Magic Squares from k-truncated Smarandache terms. {Paper presented to the FIRST INTERNATIONAL CONFERENCE ON SMARANDACHE TYPE NOTIONS IN NUMBER THEORY, University of Craiova, Romania, August 21-24, 1997} (Y. Chebrakov, V. Shmagin) Reference: [1] C.Dumitrescu & V.Seleacu, "Some Notions and Questions in Number Theory", Erhus Univ. Press, Glendale, 1994. 7) The System-Graphical Analysis of Some Numerical Smarandache Sequences An analytical investigation of 6 Smarandache Sequences of 1st kind: 1, 12, 123, 1234, 12345, 123456, ... 1, 11, 121, 1221. 12321, 123321, ... 1, 212, 32123, 4321234, 543212345, ... 1, 23, 456, 7891, 23456, 789123, 4567891, ... 1, 12, 21, 123, 231, 312, 1234, 2341, 3412, 4123, ... 12, 1342, 135642, 13578642, 13579108642, ... permitted to state that the terms of these sequences are given by the general recurrent expression: psi(a ) n a = sigma(a 10 + a + 1), where phi(n) and psi(n) are phi(n) n n functions, sigma is an operator. The main goal of the present research is to demonstrate that the system- graphical analysis results of these sequences possess big aesthetic, cognitive and applied significances. All Smarandache circumferences associated with these sequences reveal Magic properties, hence it will be very interesting to confront them with the ancient Chinese hexagrams. (Y. Chebrakov, V. Shmagin) 8) Smarandache Prime-Digital Sub-Sequence (IV) "Personal Computer World" Numbers Count of February 1997 presented some of the Smarandache Sequences and related open problems. One of them defines the Smarandache Prime-Digital Sub-Sequence as the ordered set of primes whose digits are all primes: 2, 3, 5, 7, 23, 37, 53, 73, 223, 227, 233, 257, 277, ... . We used a computer program in Ubasic to calculate the first 100 terms of the sequence. The 100-th term is 33223. Sylvester Smith [1] conjectured that the sequence is infinite. In this paper we will prove that this sequence is in fact infinite. (H. Ibstedt) Reference: [1] Smith, Sylvester, "A Set of Conjectures on Smarandache Sequences", in , India, Vol. 15E, No. 1, 1996, pp. 101-107. 9) Smarandache G Add-On Sequence (V) "Personal Computer World" Numbers Count of February 1997 presented some of the Smarandache Sequences and related open problems. Let G = {g1, g2, ..., gk, ...} be an ordered set of positive integers with a given property G. Then the corresponding Smarandache G Add-On Sequence is defined through 1+log (g ) 10 k SG = {a : a = g , a = a 10 + g , k >= 1}. i 1 1 k k-1 k I study some particular cases of this sequence, that I have presented at the FIRST INTERNATIONAL CONFERENCE ON SMARANDACHE TYPE NOTIONS IN NUMBER THEORY, University of Craiova, Romania, August 21-24, 1997. (H. Ibstedt) 10) Examples of Smarandache G Add-On Sequences (VI) The following particular cases are studied: 1) Smarandache Odd Sequence is generated by choosing G = {1, 3, 5, 7, 9, 11, ...}, and it is: 1, 13, 135, 1357, 13579, 1357911, 13571113, ... . Using the elliptic curve prime factorization program we find the first five prime numbers among the first 200 terms of this sequence, i.e. the ranks 2, 15, 27, 63, 93. But are they infinitely or finitely many? 2) Smarandache Even Sequence is generated by choosing G = {2, 4, 6, 8, 10, 12, ...}, and it is: 2, 24, 246, 2468, 246810, 24681012, ... . Searching the first 200 terms of the sequence we didn't find any n-th perfect power among them, no perfect square, nor even of the form 2p, where p is a prime or pseudo-prime. Conjecture: There is no n-th perfect power term! 3) Smarandache Prime Sequence is generated by choosing G = {2, 3, 5, 7, 11, 13, 17, ...}, and it is: 2, 23, 235, 2357, 235711, 23571113, 2357111317, ... . Terms #2 and #4 are primes; terms #128 (of 355 digits) and #174 (of 499 digits) might be, but we couldn't check -- among the first 200 terms of the sequence. Question: Are there infinitely or finitely many such primes? (H. Ibstedt) 11) Smarandache Non-Arithmetic Progression (I) "Personal Computer World" Numbers Count of February 1997 presented some of the Smarandache Sequences and related open problems. One of them defines the Smarandache t-Term Non-Arithmetic Progression as the set: {a : a is the smallest integer such that a > a , i i i i-1 and there are at most t-1 terms in an arithmetic progression}. A QBASIC program is designed to implement a strategy for building a such progression, and a table for the 65 first terms of the smarandache non-arithmetic progressions for t=3 to 15 is given. (H. Ibstedt) 12) Smarandache Prime-Product Sequence (II) "Personal Computer World" Numbers Count of February 1997 presented some of the Smarandache Sequences and related open problems. One of them defines the Smarandache Prime-Product Sequence as the set: {t : t = p # + 1, where p is the n-th prime and p # denotes n n n n n the product of all first n prime numbers}. Question: how many of them are prime? Using a computer program in QBASIC we find the first six prime numbers in this sequence: 3, 7, 31, 211, 2311, 200560490131. Are there infinitely many? The number of primes q among the first 200 terms is 6 <= q <= 9. The three terms which are either primes or pseudo primes are terms numero 75, 171, and 172 having 154, 425, and 428 digits respectively. The later two are generated by the prime twins 1019 and 1021. (H. Ibstedt) 13) Smarandache Square-Factorial Sequence (III) "Personal Computer World" Numbers Count of February 1997 presented some of the Smarandache Sequences and related open problems. One of them defines the Smarandache Square-Factorial Sequence as the set: 2 {f : f = (n!) + 1} n n We study how many terms of this sequence are prime? Among the 40 first terms we got the following primes: 2, 5, 37, 577, 14401, 131681894401, 13168189440001, 1593350922240001, 38775788043632640001, 384956219213331276939737002152967117209600000001. Are there infinitely many such primes? the product of all first n prime numbers. Question: how many of them are prime? Using a computer program in QBASIC we find six prime numbers in this sequence: 3, 7, 31, 211, 2311, 200560490131. Are there infinitely many? The number of primes q among the first 200 terms is 6 <= q <= 9. The three terms which are either primes or pseudo primes are terms numero 75, 171, and 172 having 154, 425, and 428 digits respectively. The later two are generated by the prime twins 1019 a (H. Ibstedt) 14) Smarandache Concatenation Type Sequences Let s , s , s , ..., s , ... be an infinite integer sequence 1 2 3 n (noted by S). Then the Smarandache Concatenation is defined as: ____ ______ s , s s , s s s , ... . 1 1 2 1 2 3 I search, in some particular cases, how many terms of this concatenated S-sequence belong to the initial S-sequence. 15) Smarandache Partition Type Sequences Let f be an arithmetic function, and R a k-relation among numbers. How many times can n be expressed under the form of; n = R ( f(n ), f(n ), ..., f(n )), 1 2 k for some k and n , n , ..., n such that n + n + ... + n = n ? 1 2 k 1 2 k Look at some particular cases: How many times can be n express as a sum of non-null squares (or cubes, or m-powers)? 16) A conjecture on Smarandache Anti-Symmetric Sequence In this paper we study a conjecture which states the Smarandache Anti-Symmetric Sequence: ______________ 123...n123...n, for n >= 1, has no perfect-power term. 17) On Smarandache Deducibility Theorem This theorem is defined in the Propositional Calculus of Mathematics Logic as: ___ If I--- A ___I B , for i = 1, 2, ..., n, then: i i ___ a) I--- A ^ A ^ ... ^ A ___I B ^ B ^ ... ^ B ; 1 2 n 1 2 n ___ a) I--- A v A v ... v A ___I B v B v ... v B . 1 2 n 1 2 n We study a similar theorem in the case when the logic operators "^" and "v" are replaced by Sheffer's operator. Reference: F. Smarandache, "Deducibility theorems in mathematics logics", in , seria St. Matematice, Vol. XVII, fasc. 2, 1979, pp. 163-168. We particularize S anf f to study interesting cases of this type of sequences. 18) On Smarandache Concurrent Lines If a polygon with n sides (n >= 4) is circumscribed to a circle, then there are at least three concurrent lines among the polygon's diagonals and the lines which join tangential points of two non-adjacent sides. (This is known as Smarandache Concurrent Lines, and generalizes a geometric theorem of Newton.) In this paper we try to extend this result in a three-dimensional space. Reference: F. Smarandache, "Problemes avec and sans problemes!", Ed. Somipress, Fes, Morocco, 1983, Problem & Solution # 5.36, p. 54. We particularize S anf f to study interesting cases of this type of sequences. 19) On Smarandache Cevians Theorem Let AA', BB', CC' be three concurrent cevians (lines), in the point P, in the triangle ABC. Then: PA/PA' + PB/PB' + PC/PC' >= 6, and PA PB PC BA CB AC ---- . ---- . ---- = ---- . ---- . ---- >= 8 PA' PB' PC' BA' CB' AC' (Smarandache Cevians Theorem). In this paper we extend this result for a quadrilateral. Reference: F. Smarandache, "Problemes avec and sans problemes!", Ed. Somipress, Fes, Morocco, 1983, Problems & Solutions # 5.37, p. 55, # 5.40, p. 58. We particularize S anf f to study interesting cases of this type of sequences. 20) On Smarandache Podaire Theorem Let AA', BB', CC' be the altitudes of the triangle ABC. Thus A'B'C' is the podaire triangle of the triangle ABC. Note AB = c, BC = a, CA = b, and A'B' = c', B'C' = a', C'A' = b'. Then: a'b' + b'c' + c'a' <= 1/4 (a^2 + b^2 + c^2) (Smarandache Podaire Theorem). In this paper we study this result for a quadrilateral. Reference: F. Smarandache, "Problemes avec and sans problemes!", Ed. Somipress, Fes, Morocco, 1983, Problem & Solution # 5.41, p. 59. We particularize S anf f to study interesting cases of this type of sequences. 21) On Smarandache Type Bases Considering any number wriiten in Smarandache Prime/Square/Cubic/ General Base, as written in the decimal base, we check: a) how many of them are primes? b) how many of them are perfect powers (particularly: perfect squares, or perfect cubes)? 22) Are There Finitely or Infinitely Smarandache Lucky Numbers? A number is said to be a Smarandache Lucky Number if an incorrect calculation leads to a correct result. For example, in the fraction 64/16 if the 6's are incorrectly cancelled (simplified) the result 4 is still correct. (We exclude trivial examples of the form 600/200 where non-aligned zeros are cancelled.) Is the set of all fractions, where such (or onather) incorrect calculation leads to a correct result, finite or infinite? More general: The Smarandache Lucky Method/Algorithm/Operation/etc. is said to be any incorrect method or algorithm or operation etc. which leads to a correct result. The wrong calculation should be fun, somehow similarly to the students' common mistakes, or to produce confusions or psradoxes. Can someone give an example of a Smarandache Lucky Derivation, or Integration, or Solution to a Differential Equation? (C. Ashbacher) Reference: [1] Smarandache, Florentin, "Collected Papers" (Vol. II), University of Kishinev, 1997. 22) Solved and Unsolved Problems on Pseudo-Smarandache Function A) In this note are solved the Problems: 1) Let p be a positive prime and s be an integer >= 2. Then: Z(p^s) = p^(s+1)-1, if p is even; or p^s-1, if p is odd. 2) The solution set of the diophantine equation Z(x) = 8 is {9, 12, 18, 36}. 3) For any positive integer n the diophantine equation Z(x) = n has solutions. B) Unsolved Problems: 4) The diophantine equation Z(x) = Z(x+1) has no solutions. 5) For any given positive number r there exists an integer s so that the absolute value of Z(s) - Z(s+1) is greater than r. (Where Z(n) is the Pseudo-Smarandache Function: the smallest integer m such that n evenly divides 1+2+3+...+m.) Reference: [1] Kashihara, Kenichiro, "Comments and Topics on Smarandache Notions and Problems", Erhus Univ. Press, Vail, 1996. 23) Construction of Elements of the Smarandache Square-Partial-Digital Subsequence The Smarandache Square-Partial-Digital Subsequence (SSPDS) is the sequence of square integers which admit a partition for which each segment is a square integer. An example is 506^2 = 256036, which has partition 256/0/36. C. Ashbacher showed that SSPDS is infinite by exibiting two infinite families of elements. We will extend his results by showing how to construct infinite families of elements of SSPDS containing desired patterns of digits. Unsolved Question 1: 441 belongs to SSPDS, and his square 441^2 = 194481 also belongs to SSPDS. Can an example be found of integers m, m^2, m^4 all belonging to SSPDS? Unsolved Question 2: It is relatively easy to find two consecutive squares in SSDPS, i.e. 12^2 = 144 and 13^2 = 169. Does SSDPS also contain three or more consecutive squares? What is the maximum length? (L. Widmer) 24) Perfect Powers in Smarandache Type Expressions (I) How many primes are there in the Smarandache Expression: x^y + y^x, where gcd(x, y) = 1 ? [J. Castillo & P. Castini] K. Kashihara announced that there are only finitely many numbers of the above form which are products of factorials. In this note we propose the following conjecture: Let a, b, and c three integers with ab nonzero. Then the equation: ax^y + by^x = cz^n, with x, y, n >= 2, and gcd(x, y) = 1, has finitely many solutions (x, y, z, n). And we prove some particular cases of it. 25) Products of Factorials in Smarandache Type Expressions (II) J. Castillo ["Mathematical Spectrum", Vol. 29, 1997/8, 21] asked how many primes are there in the Smarandache n-Expression: x1^x2 + x2^x3 + ... + xn^x1, where n > 1, x1, x2, ..., xn > 1, and gcd (x1, x2, ..., xn) = 1 ? [This is a generalization of the Smarandache 2-Expression: x^y + y^x.] In this note we announce a lower bound for the size of the largest prime divisor of an expression of type ax^y + by^x, where ab is nonzero, x, y >= 2, and gcd (x, y) = 1. (F. Luca) 26) The Smarandache General Periodic Sequence Definition: Let S be a finite set, and f : S ---> S be a function defined for all elements of S. There will always be a periodic sequence whenever we repeat the composition of the function f with itself more times than card(S), accordingly to the box principle of Dirichlet. [The invariant sequence is considered a periodic sequence whose period length has one term.] Thus the Smarandache General Periodic Sequence is defined as: a1 = f(s), where s is an element of S; a2 = f(a1) = f(f(s)); a3 = f(a2) = f(f(a1)) = f(f(f(s))); and so on. We particularize S anf f to study interesting cases of this type of sequences. (M. R. Popov) 27) The Two-Digit Smarandache Periodic Sequence (I) Let N1 be an integer of at most two digits and let N1' be its digital reverse. One defines the absolute value N2 = abs (N1 - N1'). And so on: N3 = abs (N2 - N2'), etc. If a number N has one digit only, one considers its reverse as Nx10 (for example: 5, which is 05, reversed will be 50). This sequence is periodic. Except the case when the two digits are equal, and the sequence becomes: N1, 0, 0, 0, ... the iteration always produces a loop of length 5, which starts on the second or the third term of the sequence, and the period is 9, 81, 63, 27, 45 or a cyclic permutation thereof. (H. Ibstedt) Reference: [1] Popov, M.R., "Smarandache's Periodic Sequences", in , University of Sheffield, U.K., Vol. 29, No. 1, 1996/7, p. 15. 28) The n-Digit Smarandache Periodic Sequence (II) Let N1 be an integer of at most n digits and let N1' be its digital reverse. One defines the absolute value N2 = abs (N1 - N1'). And so on: N3 = abs (N2 - N2'), etc. If a number N has less than n digits, one considers its reverse as N'x(10^k), where N' is the reverse of N and k is the number of missing digits, (for example: the number 24 doesn't have five digits, but can be written as 00024, and reversed will be 42000). This sequence is periodic according to Dirichlet's box principle. The Smarandache 3-Digit Periodic Sequence (domain 100 <= N1 <= 999): - there are 90 symmetric integers, 101, 111, 121, ..., for which N2 = 0; - all other initial integers iterate into various entry points of the same periodic subsequence (or a cyclic permutation thereof) of five terms: 99, 891, 693, 297, 495. The Smarandache 4-Digit Periodic Sequence (domain 1000<= N1 <= 9999): - the largest number of iterations carried out in order to reach the first member of the loop is 18, and it happens for N1 = 1019; - iterations of 8818 integers result in one of the following loops (or a cyclic permutation thereof): 2178, 6534; or 90, 810, 630, 270, 450; or 909, 8181, 6363, 2727, 4545; or 999, 8991, 6993, 2997, 4995; - the other iterations ended up in the invariant 0. (H. Ibstedt) 29) The 5-Digit and 6-Digit Smarandache Periodic Sequences (III) Let N1 be an integer of at most n digits and let N1' be its digital reverse. One defines the absolute value N2 = abs (N1 - N1'). And so on: N3 = abs (N2 - N2'), etc. If a number N has less than n digits, one considers its reverse as N'x(10^k), where N' is the reverse of N and k is the number of missing digits, (for example: the number 24 doesn't have five digits, but can be written as 00024, and reversed will be 42000). This sequence is periodic according to Dirichlet's box principle, leading to invariant or a loop. The Smarandache 5-Digit Periodic Sequence (domain 10000 <= N1 <= 99999): - there are 920 integers iterating into the invariant 0 due to symmetries; - the other ones iterate into one of the following loops (or a cyclic permutation of these): 21978, 65934; or 990, 8910, 6930, 2970, 4950; or 9009, 81081, 63063, 27027, 45045; or 9999, 89991, 69993, 29997, 49995. The Smarandache 6-Digit Periodic Sequence (domain 100000 <= N1 <= 999999): - there are 13667 integers iterating into the invariant 0 due to symmetries; - the longest sequence of iterations before arriving at the first loop member is 53 for N1 = 100720; - the loops have 2, 5, 9, or 18 terms. (H. Ibstedt) 30) The Smarandache Subtraction Periodic Sequences (IV) Let c be a positive integer. Start with a positive integer N, and let N' be its digital reverse. Put N1 = abs(N1' - c), and let N1' be its digital reverse. Put N2 = abs (N1' - c), and let N2' be its digital reverse. And so on. We shall eventually obtain a repetition. For example, with c = 1 and N = 52 we obtain the sequence: 52, 24, 41, 13, 30, 02, 19, 90, 08, 79, 96, 68, 85, 57, 74, 46, 63, 35, 52, ... . Here a repetition occurs after 18 steps, and the length of the repeating cycle is 18. First example: c = 1, 10<= N <= 999. Every other member of this interval is an entry point into one of five cyclic periodic sequences (four of these are of length 18, and one of length 9). When N is of the form 11k or 11k-1, then the iteration process results in 0. Second example: 1 <= c <= 9, 100 <= N <= 999. For c = 1, 2, or 5 all iterations result in the invariant 0 after, sometimes, a large number of iterations. For the other values of c there are only eight different possible values for the length of the loops, namely 11, 22, 33, 50, 100, 167, 189, 200. For c = 7 and N = 109 we have an example of the longest loop obtained: it has 200 elements, and the loop is closed after 286 iterations. (H. Ibstedt) 31) The Smarandache Multiplication Periodic Sequences (V) Let c > 1 be a positive integer. Start with a positive integer N, multiply each digit x of N by c and replace that digit by the last digit of cx to give N1. And so on. We shall eventually obtain a repetition. For example, with c = 7 and N = 68 we obtain the sequence: 68, 26, 42, 84, 68, ... . Integers whose digits are all equal to 5 are invariant under the given operation after one iteration. One studies the Smarandache One-Digit Multiplication Periodic Sequences only. (For c of two or more digits the problem becomes more complicated.) If c = 2, there are four term loops, starting on the first or second term. If c = 3, there are four term loops, starting with the first term. If c = 4, there are two term loops, starting on the first or second term (could be called Smarandache Switch or Pendulum). If c = 5 or 6, the sequence is invariant after one iteration. If c = 7, there are four term loops, starting with the first term. If c = 8, there are four term loops, starting with the second term. If c = 9, there are two term loops, starting with the first term (pendulum). (H. Ibstedt) 32) The Smarandache Mixed Composition Periodic Sequences (VI) Let N be a two-digit number. Add the digits, and add them again if the sum is greater than 10. Also take the absolute value of their difference. These are the first and second digits of N1. Now repeat this. For example, with N = 75 we obtain the sequence: 75, 32, 51, 64, 12, 31, 42, 62, 84, 34, 71, 86, 52, 73, 14, 53, 82, 16, 75, ... . There are no invariants in this case. Four numbers: 36, 90, 93, and 99 produce two-element loops. The longest loops have 18 elements. There also are loops of 4, 6, and 12 elements. There will always be a periodic (invariant) sequence whenever we have a function f : S ---> S, where S is a finite set, and we repeat the function f more times than card(S). Thus the Smarandache General Periodic Sequence is defined as: a1 = f(s), where s is an element of S; a2 = f(a1) = f(f(s)); a3 = f(a2) = f(f(a1)) = f(f(f(s))); and so on. (H. Ibstedt) 33) New Smarandache Sequences: The Family of Metallic Means The family of Smarandache Metallic Means (whom most prominent members are the Golden Mean, Silver Mean, Bronze Mean, Nickel Mean, Copper Mean, etc.) comprises every quadratic irrational number that is the positive solution of one of the algebraic equations 2 2 x - nx - 1 = 0 or x - x - n = 0, where n is a natural number. All of them are closely related to quasi-periodic dynamics, being therefore important basis of musical and architectural proportions. Through the analysis of their common mathematical properties, it becomes evident that they interconnect different human fields of knowledge, in the sense defined by Florentin Smarandache ("Paradoxist Mathematics"). Being irrational numbers, in applications to different scientific disciplines, they have to be approximated by ratios of integers -- which is the goal of this paper. (Vera W. de Spinadel) 34) About the Behaviour of Some New Functions in the Number Theory We investigate and prove the functions: S1 : N-{0,1} ---> N, S1(n) = 1/S(n); S2 : N* ---> N, S2(n) = S(n)/n verify the Lipschitz condition, but the functions: S3 : N-{0,1} ---> N, S3(n) = n/S(n); Fs : N* ---> N, x Fs(x) = sigma( S(p ) for i from 1 to pi(x), i where p are the prime numbers not greater than x and i pi(x) is the number of them; Theta : N* ---> N, x Theta(x) = sigma S(p ), where p are prime numbers i i which divide x; _____ Theta : N* ---> N, _____ x Theta(x) = sigma S(p ), where p are prime numbers i i which do not divide x; where S(n) is the Smarandache function for all six previous functions, verify the Lipschitz condition. (V. Seleacu, S. Zanfir)