On the F.Smarandache LCM function and its mean value

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.

<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>

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>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>

<p>For a positive integer n. the Smarandache function S(n) is defined as the smallest positive integer such that S(n)! is divisible by n.</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.

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.

L(2,1)-colorings and Irreducible No-hole Colorings of Cartesian Product of Graphs

Characterization of irreducible Boolean matrices with the largest generalized competition index

For each positive integer $n$, let $g_\Delta (n)$ be the smallest positive integer $g$ such that every complete quadratic polynomial in $n$ variables which can be represented by a sum of odd squares is represented by a sum of at most $g$ odd squares. In t

Subdivisions of digraphs in tournaments

Polynomial analogue of the Smarandache function

A bound on the generalized competition index of a primitive matrix using Boolean rank

Exponents of indecomposability

We construct a nonmetric indecomposable homogeneous continuum with the property that each of its proper subcontinua is an arc.In [5] the second author showed that in the metric case, homogeneous indecomposable continua having only arcs as proper subcontinua are solenoids.In [4] the first author defined a continuum that is indecomposable, nonmetrizable, having only arcs for proper subcontinua.Here we show that this continuum is homogeneous.This answers a question that the first author asked at the Warsaw Geometric Topology Conference in 1978.Let us start by redefining the example.Let S = {elt: t is a real number and i s= v7-1} denote the unit circle in the complex plane.If a = etr and b b= ets, then [a, b) denotes the set {c G S : c = elt and r < t < s}, and (a, b] = ([a, b)u{b})\{a).Let P = Sx{0,l} and let the topology on D be defined by pairs of half-open intervals [a, b) X {1} U (a, b] X {0}, where a, b G S. The space D is sometimes called the double arrow space.Let f:D - D be a homeomorphism defined by f(elt,j) = (elt+l,j).The quotient space K obtained from the space D X [0,1] by identification of points (d, 0) with (f(d), 1), for each d G D, is, as shown in [4], an indecomposable continuum.The same result can be obtained using theorems proved in [3].Let qf be the quotient map from V X [0,1] onto K.We will show that every proper subcontinuum of K is an arc.This argument also will give us another proof that K is indecomposable.Let M be a properbe an open set in K\M such that 0 < u < v < 1 and for some point q(p,r) of M, p does not belong to [a,b] X {0,1}.Assume without loss of generality that p G S X {1}.Let m be the smallest positive integer such that fm(p) G [a, b) X {!}-Let n be the smallest positive integer such that f~n(p) G [a, b) X {1}.Let L be the arc 1f({f-n(p)}X[0,u} U (J {{/fc(p)} X [0,1] : k == -n + 1,..., m -1} U {fm(p)} X [v, 1]) in K\U from qf(f~n(p),u) to qf(fm(p),v).Next we show that M is a subarc of L. To accomplish this let z be a point of K\(L U U).There is a closed-open set D in D such that p G D and f~n(D) (J