Related papers: On tiling the integers with $4$-sets of the same g…

Let $f(n,k)$ be the largest number of positive integers not exceeding $n$ from which one cannot select $k+1$ pairwise coprime integers, and let $E(n,k)$ be the set of positive integers which do not exceed $n$ and can be divided by at least…

We study the distributions of waiting times in variations of the negative binomial distribution of order $k$. One variation apply different enumeration scheme on the runs of successes. Another case considers binary trials for which the…

For a given subset $A\subseteq \mathbb F_q^*$, we study the problem of finding a large packing set $B$ of $A$, that is, a set $B \subseteq \mathbb F_q^*$ such that $|AB|=|A||B|$. We prove the existence of such a $B$ of size $|B|\ge…

For positive integers $n, L$ and $s$, consider the following two sets that both contain partitions of $n$ with the difference between the largest and smallest parts bounded by $L$: the first set contains partitions with smallest part $s$,…

A partition of degree $n$ is a decomposition $n=i_1+i_2+\dots+i_q$, where ${i_1,i_2,\dots,i_q}$ are positive integers called the parts of the partition. Let $\lambda>0$ be an integer. The partition is said to be a $\lambda$--partition if…

Given a sequence of $N$ positive real numbers $\{a_1,a_2,..., a_N \}$, the number partitioning problem consists of partitioning them into two sets such that the absolute value of the difference of the sums of $a_j$ over the two sets is…

Let $p$ and $q$ be two distinct fixed prime numbers and $(n_i)_{i\geq 0}$ the sequence of consecutive integers of the form $p^a\cdot q^b$ with $a,b\ge 0$. Tijdeman gave a lower bound (1973) and an upper bound (1974) for the gap size…

Let $\mathbb{F}_q$ be a finite field of order $q$. In this paper, we study the distribution of rectangles in a given set in $\mathbb{F}_q^2$. More precisely, for any $0<\delta\le 1$, we prove that there exists an integer $q_0=q_0(\delta)$…

A study of the set N_p of positive integers which occur as orders of nonsingular derivations of finite-dimensional non-nilpotent Lie algebras of characteristic p>0 was initiated by Shalev and continued by the present author. The main goal…

This paper completes the classification of maximal unrefinable partitions, extending a previous work of Aragona et al. devoted only to the case of triangular numbers. We show that the number of maximal unrefinable partitions of an integer…

Given a sequence $S=(s_1,\dots,s_m) \in [0, 1]^m$, a block $B$ of $S$ is a subsequence $B=(s_i,s_{i+1},\dots,s_j)$. The size $b$ of a block $B$ is the sum of its elements. It is proved in [1] that for each positive integer $n$, there is a…

We define a sequence of positive integers recursively, where each term is determined as follows: starting with a given positive integer, if the term is odd, the next is the sum of its positive divisors; if the term is even, the subsequent…

Let $S = \{q_1, \ldots , q_s\}$ be a finite, non-empty set of distinct prime numbers. For a non-zero integer $m$, write $m = q_1^{r_1} \ldots q_s^{r_s} M$, where $r_1, \ldots , r_s$ are non-negative integers and $M$ is an integer relatively…

Let $a=(a_1,\ldots,a_n)$ and $b=(b_1,\ldots,b_n)$ be two $n$-tuples of positive integers, let $X$ be a set of positive integers, and let $g$ be a positive integer. In this work we show an algorithmic process in order to compute all the sets…

A positive integer is called an $E_j$-number if it is the product of $j$ distinct primes. We prove that there are infinitely many triples of $E_2$-numbers within a gap size of $32$ and infinitely many triples of $E_3$-numbers within a gap…

Let $f: \mathbb{Z}_+\rightarrow \mathbb{Z}_+$ be a polynomial with the property that corresponding to every prime $p$ there exists an integer $\ell$ such that $p\nmid f(\ell)$. In this paper, we establish some equidistributed results…

A binary matrix satisfies the consecutive ones property (COP) if its columns can be permuted such that the ones in each row of the resulting matrix are consecutive. Equivalently, a family of sets F = {Q_1,..,Q_m}, where Q_i is subset of R…

The well known binary and decimal representations of the integers, and other similar number systems, admit many generalisations. Here, we investigate whether still every integer could have a finite expansion on a given integer base b, when…

We consider the problem of finding small prime gaps in various sets of integers $\mathcal{C}$. Following the work of Goldston-Pintz-Yildirim, we will consider collections of natural numbers that are well-controlled in arithmetic…

Separating hash families are useful combinatorial structures which generalize several well-studied objects in cryptography and coding theory. Let $p_t(N, q)$ denote the maximum size of universe for a $t$-perfect hash family of length $N$…