On arithmetic numbers

An integer n is said to be arithmetic if the arithmetic mean of its divisors is an integer. In this paper, using properties of the factorization of values of cyclotomic polynomials, we characterize arithmetic numbers. As an application, we give an interesting characterization of Mersenne numbers.     

On Matula numbers

Matula numbers provide a one-to-one correspondence between natural numbers and the set of rooted trees, and their significance comes from application in organic chemistry. Several results concerning Matula numbers are discussed.     

On powersmooth numbers

A natural number n is called y-smooth (y-powersmooth, respectively) for a positive number y if every prime (prime power) dividing n is bounded from above by y. Let ψ(x, y) and ψ*(x, y) denote the quantity of y-smooth and y-powersmooth integers restricted by x, respectively. In this paper we investigate function ψ*(x, y) in general. We derive formulas for finding exact calculation of ψ*(x, y) for large x and relatively small y and give theoretical estimates for this function and for a function of the greatest powersmooth integer. This results can be used in the cryptography and number theory to estimate the convergence of factorization algorithms.     

On cullen numbers

A table of Karst which gives the factorizations ofC _n =n2 ⁿ +1 is completed ton=101.     

On Delannoy numbers and Schröder numbers

The nth Delannoy number and the nth Schröder number given by

     

On the Adjacency-Jacobsthal numbers

In this article, we define the adjacency-Jacobsthal sequence and then we obtain the combinatorial representations and the sums of adjacency-Jacobsthal numbers by the aid of generating function and generating matrix of the adjacency-Jacobsthal sequence. Also, we derive the determinantal and the permanental representations of adjacency-Jacobsthal numbers by using certain matrices which are obtained from generating matrix of adjacency-Jacobsthal numbers. Furthermore, using the roots of characteristic polynomial of the adjacency-Jacobsthal sequence, we produce the Binet formula for adjacency-Jacobsthal numbers. Finally, we give the relationships between adjacency-Jacobsthal numbers and Fibonacci, Pell, and Jacobsthal numbers.     

On induced Folkman numbers

In 1970, Folkman proved that for any graph G there exists a graph H with the same clique number as G. In addition, any r ‐coloring of the vertices of H yields a monochromatic copy of G. For a given graph G and a number of colors r let f(G, r) be the order of the smallest graph H with the above properties. In this paper, we give a relatively small upper bound on f(G, r) as a function of the order of G and its clique number. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 40, 493–500, 2012     

On regular polytope numbers

Lagrange proved a theorem which states that every nonnegative integer can be written as a sum of four squares. This result can be generalized as the polygonal number theorem and the Hilbert-Waring problem. In this paper, we shall generalize Lagrange's sum of four squares theorem further. To each regular polytope in a Euclidean space, we will associate a sequence of nonnegative integers which we shall call regular polytope numbers, and consider the problem of finding the order of the set of regular polytope numbers associated to .

     

On stable crossing numbers

Results giving the exact crossing number of an infinite family of graphs on some surface are very scarce. In this paper we show the following: for G = Q_n × K_4.4, cr_y(G)-m(G) = 4m, for 0 ? = m ? 2ⁿ. A generalization is obtained, for certain repeated cartesian products of bipartite graphs. Nonorientable analogs are also developed.     

On the constructible numbers

Let be the field of constructible numbers, i.e. the numbers constructed from a given unit length using ruler and compass. We prove is definable in .

     

On Kelley's intersection numbers

We introduce a notion of weak intersection number of a collection of sets, modifying the notion of intersection number due to J.L. Kelley, and obtain an analogue of Kelley's characterization of Boolean algebras which support a finitely additive strictly positive measure. We also consider graph-theoretic reformulations of the notions of intersection number and weak intersection number.

     

On univoque Pisot numbers

We study Pisot numbers which are univoque, i.e., such that there exists only one representation of as , with . We prove in particular that there exists a smallest univoque Pisot number, which has degree . Furthermore we give the smallest limit point of the set of univoque Pisot numbers.

     

On consecutive happy numbers

Let e?1 and b?2 be integers. For a positive integer with 0?aj<b, define