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Vol. 4, No. 4, 2008ISSN 1556-6706SCIENTIA MAGNAAn international journalEdited byDepartment of MathematicsNorthwest UniversityXi’an, Shaanxi, P.R.China

Scientia Magna is published annually in 400-500 pages per volume and 1,000 copies.It is also available in microfilm format and can be ordered (online too) from:Books on DemandProQuest Information & Learning300 North Zeeb RoadP.O. Box 1346Ann Arbor, Michigan 48106-1346, USATel.: 1-800-521-0600 (Customer Service)URL: http://wwwlib.umi.com/bod/Scientia Magna is a referred journal: reviewed, indexed, cited by the followingjournals: "Zentralblatt Für Mathematik" (Germany), "Referativnyi Zhurnal" and"Matematika" (Academia Nauk, Russia), "Mathematical Reviews" (USA), "ComputingReview" (USA), Institute for Scientific Information (PA, USA), "Library of CongressSubject Headings" (USA).Price: US 69.95i

Information for AuthorsPapers in electronic form are accepted. They can be e-mailed in MicrosoftWord XP (or lower), WordPerfect 7.0 (or lower), LaTeX and PDF 6.0 or lower.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 authorsto keep a copy.The title of the paper should be writing with capital letters. The author'sname has to apply in the middle of the line, near the title. References should bementioned in the text by a number in square brackets and should be listedalphabetically. Current address followed by e-mail address should apply at the endof the paper, after the references.The paper should have at the beginning an abstract, followed by the keywords.All manuscripts are subject to anonymous review by three independentreviewers.Every letter will be answered.Each author will receive a free copy of the journal.ii

Contributing to Scientia MagnaAuthors of papers in science (mathematics, physics, philosophy, psychology,sociology, linguistics) should submit manuscripts, by email, to theEditor-in-Chief:Prof. Wenpeng ZhangDepartment of MathematicsNorthwest UniversityXi’an, Shaanxi, P.R.ChinaE-mail: wpzhang@nwu.edu.cnOr anyone of the members ofEditorial Board:Dr. W. B. Vasantha Kandasamy, Department of Mathematics, Indian Institute ofTechnology, IIT Madras, Chennai - 600 036, Tamil Nadu, India.Dr. Larissa Borissova and Dmitri Rabounski, Sirenevi boulevard 69-1-65, Moscow105484, Russia.Dr. Huaning Liu, Department of Mathematics, Northwest University, Xi’an,Shaanxi, P.R.China. E-mail: hnliu@nwu.edu.cnProf. Yuan Yi, Research Center for Basic Science, Xi’an Jiaotong University,Xi’an, Shaanxi, P.R.China.E-mail: yuanyi@mail.xjtu.edu.cnDr. Zhefeng Xu, Department of Mathematics, Northwest University, Xi’an,Shaanxi, P.R.China. E-mail: zfxu@nwu.edu.cn; zhefengxu@hotmail.comDr. Zuoren Wang, School of Statistics, Xi’an University of Finance and Economics,Xi’an, Shaanxi, P.R.China. E-mail: wangzuoren@163.comDr. Yanrong Xue, Department of Mathematics, Northwest University, Xi’an,Shaanxi, P.R.China. E-mail: xueyanrong1203@163.comiii

ContentsM. Zhu : An equation involving the Smarandache function and its positive integersolutions1C. Zhang and Y. Liu : On the Smarandache kn-digital subsequence4X. Fan : On the Pseudo-Smarandache-Squarefree function and Smarandache function 7S. Shang : On the Smarandache prime-digital subsequence sequences12L. Zhang, etc. : On the mean value of a2 (n)15X. Li : Monotonicity properties for the Gamma and Psi functions18B. Chen and C. Shi : Resolvent dynamical systems for set-valued quasivariational inclusions in Banach spaces24M. Lü, etc. : A note on a theorem of Calderón31Y. Yang : On the Smarandache sequences36S. Ren and W. He : The study of σ index on Q(Pk , Cs1 , Cs2 , · · · , Csk ) graphs40Y. Chen, etc. : The natural partial order on Semiabundant semigroups46L. Zhang, etc. : On the Smarandache ceil function and the Dirichletdivisor function55T. Veluchamy and P.S.Sivakkumar : On fuzzy number valued Choquet integral58A. A. Majumdar : The Smarandache bisymmetric arithmetic determinant sequence 62M. Lü, etc. : On the divisor function and the number of finite abeliangroups68Z. Gao, etc. : The quintic supported spline wavelets with numerical integration72A. Saeid and M. Haveshki : Approximation in Hilbert algebras77S. Balasubramanian : On ν Ti , ν Ri and ν Ci axioms86A. A. Majumdar : A note on the near pseudo Smarandache function104S. Wang, etc. : Research on the scheduling decision in fuzzy multi-resourceemergency systems112N. Wu : On the Smarandache 3n-digital sequence and the Zhang Wenpeng’sconjecture120iv

Scientia MagnaVol. 4 (2008), No. 4, 1-3An equation involving the Smarandachefunction and its positive integer solutions1Minhui ZhuSchool of Science, Xi’an Polytechnic University, Xi’an, Shaanxi, P.R.ChinaAbstract For any positive integer n 2, the Smarandache function S(n) is defined as thesmallest positive integer m such that n m!. The main purpose of this paper is using theelementary method to study the solvability of the equation S(x) n, and give an exactcalculating formula for the number of all solutions of the equation.Keywords Smarandache function, equation, positive integer solution.§1. IntroductionFor any positive integer n, the famous Smarandache function S(n) is defined as the smallestpositive integer m such that n m!. That is, S(n) min{m : n m!, m N }. From theαk1 α2definition of S(n) one can easily deduce that if n pα1 p2 · · · pk be the factorization of n intoαiprime powers, then S(n) max {S(pi )}. From this formula we can get S(1) 1, S(2) 2,1 i kS(3) 3, S(4) 4, S(5) 5, S(6) 3, S(7) 7, S(8) 4, S(9) 6, S(10) 5, S(11) 11,S(12) 4, S(13) 13, S(14) 7, S(15) 5, S(16) 6, · · · . About the other elementaryproperties of S(n), many people had studied it, and obtained a series important results, seereferences [2], [3], [4] and [5]. For example, Dr. Xu Zhefeng [4] proved the following conclusion:Let P (n) denotes the largest prime divisor of n, then for any real number x 1, we have theasymptotic formula:Ã 3 !¡ 3X2ζ 32 x 2x22 O,(S(n) P (n)) 3 ln xln2 xn xwhere ζ(s) denotes the Riemann zeta-function.Charles Ashbacher [5] studied the solvability of the equation s(m) n!, and proved thatfor any positive integer n and prime p with p n, there are some integers k such that¡ S pk n!.In this paper, we using the elementary method to study the solvability of the equationS(p ) n (where p be a prime), and give an exact calculating formula for the number of allsolutions of the equation. That is, we shall prove the following two conclusions:k1 Thiswork is supported by the Shaanxi Provincial Education Department Foundation 07JK267.

2Minhui ZhuNo. 4Theorem 1. Let n 2 be a positive integer, p be any prime with pα k n. Then there areexact α positive integers k such that the equation¡ S pk n,where pα k n denotes that pα n and pα 1 † n.Theorem 2. Let n be a fixed integer with n 2. A(k) denotes the number of all solutionsof the equation S(x) k. Then we have the calculating formula¶nXYµn β(n, p)A(k) 1 ,p 1k 1whereYp ndenotes the product over all primes p less than or equal to n, β(n, p) denotes thep nsum of the base p digits of n.It is clear that from our Theorem 1 we may immediately deduce Charles Ashbacher’s resultin reference [5]. In fact, we can get following more accurate result:Corollary. Let n be a positive integer, then for any prime p with p n, there are exactn β(n, p)p 1¡ integers k such that the equation S pk n!, where β(n, p) is defined as in Theorem 2.Of course, our Corollary is also holds if p n. In this case, β(n, p) n, so that¡ n β(n, p) 0. Therefore, the equation S pk n! has no positive integer solution.p 1§2. Proof of the theoremsIn this section, we shall use the elementary method to complete the proof of our Theorems.First we prove Theorem 1. Let n 2 be an integer, for any prime p with pα k n, if α 0,¡ then (p, n) 1, and the equation S pk n has no positive integer solution k. Otherwise,p n, this contradiction with (p, n) 1. So the number of all positive integer solutions of the¡ equation S pk n is α 0. That is means, Theorem 1 is true if α 0. Now we assume thatα 1, and let pα(n,p) k n!. Then from the elementary number theory textbook (see [6], [7])and reference [8] we know that ·Xn β(n, p)nα(n, p) ,(1)ipp 1i 1where β(n, p) denotes the sum of the base p digits of n. That is, if n a1 pα1 a2 pα2 · · · as pαssXwith αs αs 1 · · · α1 0 and 1 ai p 1, i 1, 2, · · · , s, then β(n, p) αi .i 1Now for all positive integers k α(n, p) α 1, α(n, p) α 2, · · · , α(n, p) 1 and¡ α(n, p), we have S pk n. Since this time, from (1) we know that pk n!, but pk † (n 1)!,because pα(n,p) α k (n 1)!. From the definition of the Smarandache function S(n) we knowthat if k α(n, p) α, then pk (n 1)!, if k α(n, p), then pk † n!. Therefore, the equation¡ S pk n has exact α positive integer solutions. This proves Theorem 1.

Vol. 4An equation involving the Smarandache function and its positive integer solutions3Now we prove Theorem 2. For any positive k 2, if x satisfy the equation S(x) k, thenx k! and x†(k 1)!. So the number of all solutions of the equation S(x) k is equal toXX1 1 d(k!) d((k 1)!),d k!d (k 1)!where d(m) is the Dirichlet divisor function. Therefore,XXXA(k) A(1) A(k) 1 [d(k!) d((k 1)!] d(n!).k n2 k n2 k nNote that (1), from the definition and properties of d(n) we may immediately get¶XYµn β(n, p)A(k) d(n!) 1 ,p 1k nwhereYp ndenotes the product over all primes p less than or equal to n, β(n, p) denotes thep nsum of the base p digits of n.This completes the proof of Theorem 2.References[1] F. Smarandache, Only Problem, Not Solutions, Chicago, Xiquan Publishing House,1993.[2] F. Smarandache, Sequences of Numbers Involved in Unsolved Problems, Hexis, 2006.[3] V. Mladen and T. Krassimir, Remarks on some of Smarandache’s problem, part 2,Scientia Magna, 2(2005), No.1, 1-26.[4] Xu Zhefeng, The value distribution property of the Smarandache function, Acta Mathematica Sinica, Chinese Series, 49(2006), No.5, 1009-1012.[5] Charles Ashbacher, Some problems on Smarandache function, Smarandache NotionsJournal, 6(1995), No.1-2-3, 21-36.[6] Tom M.Apostol, Introduction to Analytic Number Theory, New York, Springer-Verlag,1976.[7] Zhang Wenpeng, The elementary number theory (in Chinese), Shaanxi Normal University Press, Xi’an, 2007.[8] Liu Hongyan and Zhang Wenpeng, A number theoretic function and its mean valueproperty, Smarandache Notions Journal, 13(2002), No.1-2-3, 155-159.[9] Wu Qibin, A composite function involving the Smarandache function, Pure and AppliedMathematics, 23(2007), No.4, 463-466.[10] Yi Yuan and Kang Xiaoyu, Research on Smarandache Problems (in Chinese), HighAmerican Press, 2006.[11] Chen Guohui, New Progress On Smarandache Problems (in Chinese), High AmericanPress, 2007.[12] Liu Yanni, Li Ling and Liu Baoli, Smarandache Unsolved Problems and New Progress(in Chinese), High American Press, 2008.

Scientia MagnaVol. 4 (2008), No. 4, 4-6On the Smarandache kn-digital subsequenceCuncao Zhang† and Yanyan Liu‡† The Primary School Attached to Northwest University, Xi’an, Shaanxi, P.R.China‡ Department of Mathematics, Northwest University, Xi’an, Shaanxi, P.R.ChinaAbstract For any positive integer n and any fixed positive integer k 2, the Smarandachekn-digital subsequence {Sk (n)} is defined as the numbers Sk (n), which can be partitionedinto two groups such that the second is k times bigger than the first. The main purpose ofthis paper is using the elementary method to study the convergent properties of the infiniteseries involving the Smarandache kn-digital subsequence {Sk (n)} , and obtain some interestingconclusions.Keywords The Smarandache kn-digital sequence, infinite series, convergence.§1. IntroductionFor any positive integer n and any fixed positive integer k 2, the Smarandache kn-digitalsubsequence {Sk (n)} is defined as the numbers Sk (n), which can be partitioned into two groupssuch that the second is k times bigger than the first. For example, the Smarandache 3n-digitalsubsequence are: S3 (1) 13, S3 (2) 26, S3 (3) 39, S3 (4) 412, S3 (5) 515, S3 (6) 618,S3 (7) 721, S3 (8) 824, S3 (9) 927, S3 (10) 1030, S3 (11) 1133, S3 (12) 1236,S3 (13) 1339, S3 (14) 1442, S3 (15) 1545, S3 (16) 1648, S3 (17) 1751, S3 (18) 1854,S3 (19) 1957, S3 (20) 2060, S3 (21) 2163, S3 (22) 2266, · · · .The Smarandache 4n-digital subsequence are: S4 (1) 14, S4 (2) 28, S4 (3) 312,S4 (4) 416, S4 (5) 520, S4 (6) 624, S4 (7) 728, S4 (8) 832, S4 (9) 936, S4 (10) 1040,S4 (11) 1144, S4 (12) 1248, S4 (13) 1352, S4 (14) 1456, S4 (15) 1560, · · · .The Smarandache 5n-digital subsequence are: S5 (1) 15, S5 (2) 210, S5 (3) 315,S5 (4) 420, S5 (5) 525, S5 (6) 630, S5 (7) 735, S5 (8) 840, S5 (9) 945, S5 (10) 1050,S5 (11) 1155, S5 (12) 1260, S5 (13) 1365, S5 (14) 1470, S5 (15) 1575, · · · .These subsequences are proposed by Professor F.Smarandache, he also asked us to studythe properties of these subsequences. About these problems, it seems that none had studiedthem, at least we have not seen any related papers before. The main purpose of this paperis using the elementary method to study the convergent properties of one kind infinite seriesinvolving the Smarandache kn-digital subsequence, and prove the following conclusion:1Theorem. Let z be a real number. If z , then the infinite series2f (z, k) X1z (n)Sn 1 k(1)

Vol. 45On the Smarandache kn-digital subsequence1is convergent; If z , then the infinite series (1) is divergent.2In these Smarandache kn-digital subsequences, it is very hard to find a complete squarenumber. So we believe that the following conclusion is correct:Conjecture. There does not exist any complete square number in the Smarandachekn-digital subsequence, where k 3, 4, 5. That is, for any positive integer m, m2 / {Sk (n)},where k 3, 4, 5.§2. Proof of the theoremIn this section, we shall use the elementary method to complete the proof of our Theorem.We just prove that the theorem is holds for Smarandache 3n-digital subsequence. Similarly, wecan deduce that the theorem is also holds for any other positive integer k 4. For any elementS3 (a) in {S3 (n)}, let 3a bk bk 1 · · · b2 b1 , where 1 bk 9, 0 bi 9, i 1, 2, · · · , k 1.Then from the definition of the Smarandache 3n-digital subsequence we have¡ S3 (a) a · 10k 3 · a a · 10k 3 .(2)On the other hand, let a as as 1 · · · a2 a1 , where 1 as 9, 0 ai 9, i 1, 2, · · · , s 1.It is clear that if a 33· · 33}, then s k; If a 33 ·{z· · 34}, then s k 1. So from the definition ·{zsof S3 (a) and the relationship of s and k we havef (z, 3) s3333333333333XXXXX 111111 ···S z (n)S z (i) i 4 S3z (i) i 34 S3z (i) i 34 S3z (i) i 334 S3z (i)n 1 3i 1 3 X3Xi 1 X333333333XXX1111 ···zzzzzzz · 10003zi · 13i·103i·1003ii 4i 34i 334 XX3 · 10k 110k1 3· 3·.z(k 2)zkz(k 1)z(k 1)k·(2z 1)10· 1010· 1010k 1k 0k 0(3)1Now if z , then from (3) and the properties of the geometric progression we know that2f (z, 3) is convergent.1If z , then from (3) we also have2f (z, 3) X3333333333333XXXXX 111111 ···S z (n)S z (i) i 4 S3z (i) i 34 S3z (i) i 34 S3z (i) i 334 S3z (i)n 1 3i 1 33Xi 1 X333333333XXX1111 ···zzzzzzz · 10003zi · 13i·103i·1003ii 4i 34i 334 XX3 · 10k 110k1 3· 3·.z(k 1)z(k 1)zkz(k 2)(2zk 2z k)10· 1010 · 1010k 1k 0k 0(4)Then from the properties of the geometric progression and (4) we know that the series f (z, 3)1is divergent if z . This proves our theorem for k 3.2

6Cuncao Zhang and Yanyan LiuNo. 4Similarly, we can deduce the other cases. For example, if k 4, then we havef (z, 4) 224249249924999XXXX 1X11111 zzzzzz (i) · · ·S(n)S(i)S(i)S(i)S(i)Sn 1 4i 1 4i 3 4i 25 4i 250 4i 2500 3 X2Xi 1242492499XXX1111 ···zzzzzzzi · 14i · 104i · 1004i · 10004zi 3i 25i 250 X XX225 · 10k 210k1 225 · 225 ·z(k 2) · 10zkzk · 10z(k 2)k·(2z 1) 2z101010k 1k 0k 0(5)andf (z, 4) X224249249924999XXXX 1X11111 zzzzzz (i) · · ·S(n)S(i)S(i)S(i)S(i)Sn 1 4i 1 4i 3 4i 25 4i 250 4i 2500 32Xi 1 242492499XXX1111 ···zzzzzzzi · 14i · 104i · 1004i · 10004zi 3i 25i 250 XX10k12 · 10k 20· 20·.z(k 1)zkzkz(k 1)k·(2z 1) z10· 1010 · 1010k 0k 0k 1 X(6)From (5), (6) and the properties of the geometric progression we know that the theorem is holdsfor the Smarandache 4n-digital subsequence.This completes the proof of Theorem.References[1] F.Smarandache, Only Problem, Not Solutions, Chicago, Xiquan Publishing House, 1993.[2] F.Smarandache, Sequences of Numbers Involved in Unsolved Problems, Hexis, 2006.[3] Yi Yuan and Kang Xiaoyu, Research on Smarandache Problems, High American Press,2006.[4] Chen Guohui, New Progress On Smarandache Problems, High American Press, 2007.[5] Liu Yanni, Li Ling and Liu Baoli, Smarandache Unsolved Problems and New Progress,High American Press, 2008.[6] Wang Yu, Su Juanli and Zhang Jin, On the Smarandache notions and related problems,High American Press, 2008.[7] Zhang Wenpeng, The elementary number theory (in Chinese), Shaanxi Normal University Press, Xi’an, 2007.[8] Tom M.Apostol, Introduction to Analytic Number Theory, New York, Springer-Verlag,1976.

Scientia MagnaVol.4 (2008), No. 4, 7-11On the Pseudo-Smarandache-Squarefreefunction and Smarandache functionXuhui Fan†Foundation Department, Engineering College of Armed Police Force,Xi’an, Shaanxi, 710086‡ Department of Mathematics, Northwest University, Xi’an, Shaanxi, 710069Abstract For any positive integer n, Pseudo-Smarandache- Squarefree function Zw (n) isdefined as Zw (n) min{m : n mn , m N }. Smarandache function S(n) is defined asS(n) min{m : n m!, m N }. The main purpose of this paper is using the elementarymethods to study the mean value properties of the Pseudo-Smarandache-Squarefree functionand Smarandache function, and give two sharper asymptotic formulas for it.Keywords Pseudo-Smarandache-Squarefree functionZw (n), Smarandache function S(n), mean value, asymptotic formula.§1. Introduction and resultFor any positive integer n, the famous Smarandache function S(n) is defined as S(n) min{m : n m!, m N }, Pseudo-Smarandache-Squarefree function Zw (n) is defined as thesmallest positive integer m such that n mn . That is,Zw (n) min{m : n mn , m N }.For example Zw (1) 1, Zw (2) 2, Zw (3) 3, Zw (4) 2, Zw (5) 5, Zw (6) 6, Zw (7) 7,Zw (8) 2, Zw (9) 3, Zw (10) 10, · · · . About the elementary properties of Zw (n), someauthors had studied it, and obtained some interesting results. For example, Felice Russo [1]obtained some elementary properties of Zw (n) as follows:Property 1. For any positive integer k 1 and prime p, we have Zw (pk ) p.Property 2. For any positive integer n, we have Zw (n) n.Property 3. The function Zw (n) is multiplicative. That is, if GCD(m, n) 1, then Zw (m·n) Zw (m) · Zw (n).The main purpose of this paper is using the elementary methods to study the mean valueproperties of Zw (S(n)) and S(n) · Zw (n), and give two sharper asymptotic formulas for it. Thatis, we shall prove the following conclusions:Theorem 1. Let k 2 be any fixed positive integer. Then for any real number x 2,

8Xuhui FanNo. 4we have the asymptotic formulaX³ x2 X ci · x2π 2 x2· O,12 ln x i 2 lni xlnk 1 xkZw (S(n)) n xwhere ci (i 2, 3 · · · k) are computable constants.Theorem 2. Let k 2 be any fixed positive integer. Then for any real number x 2,we have the asymptotic formulaX³ x3 X ei · x3ζ(2) · ζ(3) Y ³1 x31 · O,3ζ(4)p p3ln x i 2 lni xlnk 1 xpkZw (n) · S(n) n xwhere ζ(n) is the Riemann zeta-function,Ydenotes the product over all primes,

S. Balasubramanian : On ” ¡Ti¡;” ¡Ri¡ and ” ¡Ci¡ axioms 86 A. A. Majumdar : A note on the near pseudo Smarandache function 104 S. Wang, etc. : Research on the scheduling decision in fuzzy multi-resource emergency systems 112 N. Wu : On the Smarandache 3n-digital sequence a

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