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 cullen numbers

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

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 Delannoy numbers and Schröder numbers

The nth Delannoy number and the nth Schröder number given by

     

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 consecutive happy numbers

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

     

On generalised catalan numbers

The Catalan number C_n is defined to be $\text{2n}\text{n}\text{(n+1)}$. One of its occurrences is as the number of ways of bracketing a product of n+1 terms taken from a set with binary operation. In this note the corresponding result for a set with a k-ary operation is considered. A combinatorial proof is given which does not involve generating functions or inversion formulae. The result is further generalised to obtain a simpler proof of a formula of Erdelyi and Etherington [2], interpreted here as a result concerning a set with several k_i-ary operations.     

On the Łojasiewicz numbers

Let f be a holomorphic function of two complex variables with an isolated critical point at $\text{0∈}\text{C}{}^2$. We give some necessary conditions for a rational number to be the smallest θ>0 in the ?ojasiewicz inequality |gradf(z)|?C|z|^θ for z near $\text{0∈}\text{C}{}^2$. To cite this article: E. Garc??a Barroso, A. P?oski, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

On Ramsey numbers of fans

It is shown that r(F₂,F_n)=4n+1 for n≥2, and r(F_s,F_n)≤4n+2s for n≥s≥2.     

On ramsey numbers for books

For n = 1, 2, …, let B_n = K₂ + K?_n. We pose the problem of determining the Ramsey numbers r(B_m, B_n) and demonstrate that in many cases critical colorings are avialable from known examples of strongly regular graphs.