ON RADU' S PROBLEM

Polynomial analogue of the Smarandache function

For any positive integer n, the F.Smarandache LCM function SL(n) is deflned as the smallest positive integer k such that n j (1;2;¢¢¢ ;k), where (1;2;¢¢¢ ;k) denotes the least common multiple of 1;2;¢¢¢ ;k, and let n = p fi 1 1 p fi2 2 ¢¢¢p fis s be the factorization of n into prime powers, then ›(n) = fi1p1 + fi2p2 + ¢¢¢ + fisps. The main purpose of this paper is using the elementary methods to study the mean value properties of ›(n)SL(n), and give a sharper asymptotic formula for it. Keywords F.Smarandache LCM function, ›(n) function, mean value, asymptotic formula.

The Ramsey number R(F) of a graph F without isolated vertices is the smallest positive integer n such that every red–blue coloring of Kn produces a subgraph isomorphic to F all of whose edges are colored the same. Let F be a set of graphs without isolated vertices. For a positive integer t, the vertex-disjoint Ramsey number VRt(F) is the smallest positive integer n such that every red–blue coloring of the complete graph Kn of order n results in at least t pairwise vertex-disjoint monochromatic graphs in F; while the edge-disjoint Ramsey number ERt(F) is the smallest positive integer n such that every red–blue coloring of Kn produces at least t pairwise edge-disjoint monochromatic graphs in F. If t=1 and F consists of a single graph F, then VR1(F)=ER1(F)=R(F) is the Ramsey number of the graph F. Thus, the concepts of vertex-disjoint and edge-disjoint Ramsey numbers provide a generalization of the standard Ramsey number. Upper and lower bounds for VRt(F) and ERt(F) are established for sets F of graphs without isolated vertices and the sharpness of these bounds is discussed. The primary goal of this paper is to investigate the values of VRt(F) and ERt(F) for sets F of graphs of size 2 or 3 without isolated vertices. The exact values of VRt(F) are determined for all such sets F and all integers t≥2. The exact values of ERt(F) of certain such sets F with prescribed conditions for all integers t≥2 are determined. For some special sets F of graphs of size 2 or 3 without isolated vertices, the exact values of ERt(F) are determined for 2≤t≤4. Additional results, problems, and conjectures are also presented dealing with these two Ramsey concepts for graphs in general.

<p>The main purpose of this paper is to use the elementary methods to study the mean value of the F.Smarandache LCM function SL(n), and give a sharper asymptotic formula for it.</p>

<p>For any positive integer n, the famous F.Smarandache LCM function SL(n) is<br /> de¯ned as the smallest positive integer k.</p>

<p>For any positive integer n and prime p, let Sp(n) denotes the smallest positive<br /> integer m such that m! is divisible by pn.</p>

Let k be a positive integer and G = (V(G), E(G)) be a finite and connected graph. A rainbow vertex k-coloring of G is a function c: V(G) → {1,2,…, k} such that for every two vertices u and v in V(G) there exists a u-v path whose internal vertices have distinct colors. Such path is called a rainbow vertex path. The rainbow vertex connection number of G, denoted by rvc(G). is the smallest positive integer k so that G has a rainbow vertex k-coloring. The distance between two difference vertices u and v in V(G), denoted by d(u,v), is the length of a shortest u-v path in G. For i ∈ {1,2,…, k], let Ri be the set of vertices with color i and Π - {R 1,R 2,…,Rk } be fjn ordered partition of V(G). The rainbow code of a vertex v of V(G) with respect to Π is defined as the k-tuple , where d(v, Ri ) =min{d(v, y) | y ∈ Ri } for each i ∈ {1,2,…, k}. If every vertex of G has distinct rainbow codes, then c is called a locating rainbow k-coloring of G. The locating rainbow connection number of G. denoted by rvcl(G). is defined as the smallest positive integer k such that G has a locating rainbow k-coloring. In this paper, we provide the sharp upper and lower bounds for locating rainbow connection number of a graph. We also determine the locating rainbow connection number of some well-known classes of graphs.

This study investigates the positive integer solutions of an equation involving the Smarandache function. The equation is given by tφ(n) = φ 2 (n) + S(n 16), where φ(n) represents the Euler function, φ e (n) represents the generalized Euler function with e as a positive integer, and S(n) represents the Smarandache function. The solutions of this equation are discussed, and it is proven that the equation only has positive integer solutions when t = 1, 2, 3, 6, 7, 9, 12, 17, 18, 19. Furthermore, all positive integer solutions of the equation are provided.

In discussing unique factorization and ideal theory, C. C. MacDuffee [l, p. 122] cites the multiplicative system of positive integers of the form 1+7& as an example where unique factorization fails, since 792 = 22-36 = 8-99, and 8, 22, 36, 99 are all primes in the system. H. Davenport [2, p. 21] uses positive integers of the form l+ik for the same purpose, with the numerical case 693 = 9-77 = 21-33. In this paper we examine all multiplicative systems made up of arithmetic progressions, and decide the question of unique factorization. For a fixed positive integer ra, let M be a multiplicatively closed system of positive integers such that if xGA2 and y=x (mod ra), y>0, then yG.M. It will be assumed that w is the smallest positive integer which can be used to define M. For example the set M of all positive integers congruent to 1, 3, or 5 modulo 6 is also the set congruent to 1 modulo 2, and in this case ra = 2. We divide the integers 1, 2, • • • , w into two classes: the set A, 0(ra) in number, of those relatively prime to ra, and the others in set B, ra—0(ra) in number.

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