The density of primes dividing at least one term of the Lucas sequence ${\left\{L_n\left(P\right)\right\}}_{n=0}^{\infty }$, defined by $L_0\left(P\right)=2,L_1\left(P\right)=P$ and $L_n\left(P\right)=P{L}_{n-1}\left(P\right)+L_{n-2}\left(P\right)$ for $n\ge 2$, with $P$ an arbitrary integer, is determined.

In [11] and [7], the concepts of σ-core and statistical core of a bounded number sequence x have been introduced and also some inequalities which are analogues of Knopp’s core theorem have been proved. In this paper, we characterize the matrices of the class ${(S\cap m,V_{\sigma })}_{reg}$ and determine necessary and sufficient conditions for a matrix A to satisfy σ-core(Ax) ⊆ st-core(x) for all x ∈ m.

The continuity of densities given by the weight functions $n^{\alpha }$, $\alpha \in [-1,\infty [$, with respect to the parameter $\alpha$ is investigated.

We are interested in permutations preserving certain distribution properties of sequences. In particular we consider $\mu$-uniformly distributed sequences on a compact metric space $X$, 0-1 sequences with densities, and Cesàro summable bounded sequences. It is shown that the maximal subgroups, respectively subsemigroups, of $Aut\left(\mathbf{N}\right)$ leaving any of the above spaces invariant coincide. A subgroup of these permutation groups, which can be determined explicitly, is the Lévy group $𝒢$. We show that $𝒢$ is big in the...

Let $S$ be an abelian semigroup, and $A$ a finite subset of $S$. The sumset $hA$ consists of all sums of $h$ elements of $A$, with repetitions allowed. Let $\left|hA\right|$ denote the cardinality of $hA$. Elementary lattice point arguments are used to prove that an arbitrary abelian semigroup has polynomial growth, that is, there exists a polynomial $p\left(t\right)$ such that $\left|hA\right|=p\left(h\right)$ for all sufficiently large $h$. Lattice point counting is also used to prove that sumsets of the form $h_1{A}_1+\cdots +h_r{A}_r$ have multivariate polynomial growth.

For any partition of a set of squarefree numbers with relative density greater than 3/4 into two parts, at least one part contains three numbers whose product is a square. Also generalizations to partitions into more than two parts are discussed.

We investigate some properties of density measures – finitely additive measures on the set of natural numbers $\mathbb{N}$ extending asymptotic density. We introduce a class of density measures, which is defined using cluster points of the sequence $\left(\frac{A\left(n\right)}{n}\right)$ as well as cluster points of some other similar sequences. We obtain range of possible values of density measures for any subset of $\mathbb{N}$. Our description of this range simplifies the description of Bhashkara Rao and Bhashkara Rao [Bhaskara Rao, K. P. S., Bhaskara Rao,...

We give an extension of Benford's law (first digit problem) by using the concept of conditional density, introduced by Fuchs and Letta. The main tool is the notion of regular subset of integers.

This paper is closely related to an earlier paper of the author and W. Narkiewicz (cf. [7]) and to some papers concerning ratio sets of positive integers (cf. [4], [5], [12], [13], [14]). The paper contains some new results completing results of the mentioned papers. Among other things a characterization of the Steinhaus property of sets of positive integers is given here by using the concept of ratio sets of positive integers.