The Smarandache function and its principal properties are already known in the literature of speciality. Other functions were built analogously, among which the following ones

Radescu, E.

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Some Elementary Algebraic Considerations Inspired by Smarandache Type Functions (II) E. Radescu "University of Craiova: Department of :\Iathematics: 1100 Craiova: Romania Abstract The paper presents new properties for some functions constructed sim-ilarly to the functiol} ry :" N* -- N* , the Smarandache function, defined by "In E N*, ry(n) = min{kJk! is divisible by n} - "Smarandache's type func-tion" " The Smarandache "I function and its principal properties are al-ready known in the literature of speciality. Other functions were built analogously, among which the following ones. The function TJI. Starting from a sequence of positive integers a : N* -- N* satisfying the condition "In E N*, 3mn E N*, "1m ~ mn =:} n/a(m) (1) an associated function was built "II : N* -- N* , defined by TJI(n) = min {mnlmn is given by (1) }, "In E N* . (2) Such sequences - possibly satisfying an extra condition - considered by G. Christol to generalise the p-adic numbers were called also mul-tiplicative convergent to zero (m.c.z.). An example is a : N* -- N* with a(n) = n!. For n = 6, there is m~ = 4 such that "1m ~ 4 =:} 6/m! ( 6/4! for m = 4; 6/5! for m = 5) but there is and m~ = 7 such that "1m ~ 7 =:} 6/m!; because the smallest of them is ms = 3 such that "1m ~ 3 =:} 6/3!, it results TJI(6) = 3. 124 We note that for 0"( n) = n! the associated function 1J1 is just the TJ function - from where the ideea of building the TJi functions (by generalization of the sequence). The function TJ2. A sequence of positive integers 0" : N* ---- N* is called "of divisibility sequence (d.s.)'"' if: min ~ O"(m)/O"(n), (3) and "of strong divisibility sequence (s.d.s.)" if O"((m,n)) = (O"(m),O"(n)), 'im,n E N-, (4) (m, n) being the greatest common factor. (Strong divisibility sequences are studied for instance by N. Jensen in [5]. It is known that the Fibonacci sequence is a s.d.s.). Starting from a sequence 0" : N* -> N* satisfying the condition 'in E N*,3mn E N-,'im E N*, mn/m ~ n/O"(m) (5) an associated function was built that is TJ2 : N* ..... N* defined by TJ2(n) = min {mnlmn is given by (5)}-, 'in EN-. (6) • If the sequence 0" is d.s. or s.d.s., the function TJ2 has new properties with interesting algebraic interpretations. We observe that in (1) appeared both the natural order (m ~ m n ) and the divisibility as relation of order on N* (n/O"(m)) and in (5), only the divisibility as relation of order on N*. From the alternation of the two relations of order on N* can be defined analogously two more functions 1/3 and TJ4. (see [1]) Starting from a sequence 0" : N* --+ N* satisfying the condition 'in E N*,3mn E N*,'im E N-,mn/m ~ n::; O"(m) (7) an associated function was built that is 1/3 : N- -> N- , defined by TJ3(n) = min {mnlmn is given by (7) }, 'in EN- . (8) Also, starting from a sequence 0" : N- ..... N- satisfying the condition 'in E N-,3mn E N-,'im E N*,mn::; m ~ n::; O"(m) (9) an associated function was built that is TJ4 : N- ..... N* , defined by TJ4(n) = min {mnlmn is given by (9)}, 'in EN-. (10) The principal properties of the functions above are divided in three groups: 125 I The arithmetical properties of the proper function. II The properties of sumatory function associated to each of the nu-merical functions above. (see [3]) III The algebraical properties of the proper function. Thanks to the arithmetical properties, every function can be viewed as mor-phism (endomorphism) between certain universal algebras (we can be obtain several situations considering various operations ofN*). (see [2], [4]) This paper presents a construction from group III which guides to a prolongation S4 of the function 174 for more complexe universal algebras. If the initial sequence is s.d.s., the associated function 174 has a series of important properties from which we retain: 174 (max{ a, b}) = max {174( a), 174 ( b)} ; (ll) 174 (min{a,b}) = min {174(a),174(b)}Va, b E N*. (12) We may stand out, from other possible structures on N* , the uni-versal algebra (N*, n) where the set of operati.ons is n = {v,!\, ?Po} with V,!\ : (N*)2 -+ N* defined by a V b· = sup{ a, b}, a !\ b = inf {a, b}, Va, b E N* (N* is a lattice with the natural order) and ?Po : (N*)o -+ N* - a null operation that fixes 1, the unique particular element with the role of neutral element for "v ": 1 = ev . Therefore, the universal algebra (N*, n) is of type r = (~ ~ ~o) = (2,2,0). With the properties (ll) and (12) the function 174 is endomorphism for the universal algebra above. It can be stated Teorema 1 If 174 : N* -+ N* is the fu.nction defined by (10), endo-morphism for the universal algebra (N*, n) and I is a set, then there is a unique S4 : (N*)I --.. (N* / , endomorphism for the universal algebra ((N*/ , n) so that Pi 0 S4 = 174 0 Pi, Vi E I, where Pj : (N*)! -+ N* with Va = {a;}iEI E (N*/, pj(a) = aj, Vj E I, are the canonical projections, morphisms between ((N*)! ,n) and (N*,n). The proof can be done directly: it is shown that the correspon-dence 174 is a function, endomorphism and complies with the required 126 conditions. The operations of n for the universal algebra ((N*/ ,n ) are made "on components". The algebraic properties of S4 - the prolongation to more ampler universal algebra of the function 1]4 - for its restriction to N* , could bring new properties for the function 1]4 that we considered above. The paper contents, in completion, a formula of calcul for the sumatory function F~ of function 172 . If the initial sequence is s.d.s., this formula is: It It F'12 (n) = 1]2(1) + L [1]2 (Ph) , 1]2(Pt)] + L [1]2(Ph), 1J2(Pt), 1]2(Pq)] + h,t=1 h# h,t,q=1 h##q + ... + 1]2 ( n), 'in = PI . P2 ... Pit, Pi, i = 1, k - prime numbers and 1J2(pa) = F'12(pa) - F'12(pa-I). References [lJ Dumitrescu, C., RoC§oreanu, C., Some connection between the Smarandache fu.nction and the Fibonacci sequence, Smarandache Notions Journal, Vol. 10, No.1-2-3, Spring, 1995, p.50-59 [2J Radescu, E., Radescu, N., Dumitrescu,' C., Some elementary algebraic considerations inspired by the Smarandache function, Smarandache Founction Journal, Vol.6,No.1.1995,p.50-54 [3J Radescu, E., Radescu, N., Some considerations concerning the sumatory function associated to Smarandache function, Smaran-dache Notions Journal, Vo1.7,No.1-2-3,1996,p.63-69 [4J Radescu, E., Radescu, N., Some elementary algebraic considera-tions inspired by Smarandache's function (II), Smarandache Na-tions Journal, Vo1.7,No.1-2-3,1996,p.70-77 [5J Jensen N., "Some Results on Divisibility Sequences", Appl. of Fi-bonacci Numbers, Vol. 3, G.E. Bergum and al. Editors, Kluwer Academic Press, 1990, pp. 181-189. 127 

Some connection between the Smarandache fu.nction and the Fibonacci sequence,

Some Elementary Algebraic Considerations Inspired by Smarandache Type Functions (II)

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Some Elementary Algebraic Considerations Inspired by Smarandache Type Functions (II)

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