Related papers: Parry order of Parry numbers

A beta expansion is the analogue of the base 10 representation of a real number, where the base may be a non-integer. Although the greedy beta expansion of 1 using a non-integer base is in general infinitely long and non-repeating, it is…

We define a partial order $\mathcal{P}_n$ on permutations of any given size $n$, which is the image of a natural partial order on inversion sequences. We call this the ``middle order''. We demonstrate that the poset $\mathcal{P}_n$ refines…

An asymptotic formula for the number of $n \le x$ such that $n$ does not divide $P(n)!$ is given, where P(n) is the largest prime factor of $n$.

We give an algorithm to enumerate all primitive abundant numbers (briefly, PANs) with a fixed $\Omega$ (the number of prime factors counted with their multiplicity), and explicitly find all PANs up to $\Omega=6$, count all PANs and…

In this article, we investigate the $\beta$-expansions of real algebraic numbers. In particular, we give new lower bounds for the number of digit exchanges in the case where $\beta$ is a Pisot or Salem number. Moreover, we define a new…

Let $r\geq3$ be an integer such that $r-2$ is a prime power and let $H$ be a connected graph on $n$ vertices with average degree at least $d$ and $\alpha(H)\leq\beta n$, where $0<\beta<1$ is a constant. We prove that the size Ramsey number…

A natural number $n$ is called semi-prime if it is a product of two primes or a square of a prime. We denote $\mathbb{P}_2$ the set of all semi-primes. Our goal is to prove that for fixed integer number $a$ and sufficiently large $x$ the…

Much has been written about expansions of real numbers in noninteger bases. Particularly, for a finite alphabet $\{0,1,\dots,\alpha\}$ and a real number (base) $1<\beta<\alpha+1$, the so-called {\em univoque set} of numbers which have a…

B\"uchi arithmetics $\mathop{\mathbf{BA}}\nolimits_n$, $n\ge 2$, are extensions of Presburger arithmetic with an unary functional symbol $V_n(x)$ denoting the largest power of $n$ that divides $x$. A rank of a linear order is the minimal…

Let ${\mathcal{P}_{n}}$ denote the set of positive integers which are prime to $n$. Let $B_{n}$ be the $n$-th Bernoulli number. For any prime $p \ge 11$ and integer $r\ge 2$, we prove that $$ \sum\limits_{\begin{smallmatrix}…

We study the problem of determining the size of the largest intersecting $P$-free family for a given partially ordered set (poset) $P$. In particular, we find the exact size of the largest intersecting $B$-free family where $B$ is the…

For $x\ge0$ let $\pi(x)$ be the number of primes not exceeding $x$. The asymptotic behaviors of the prime-counting function $\pi(x)$ and the $n$-th prime $p_n$ have been studied intensively in analytic number theory. Surprisingly, we find…

Let $I(t)= \oint_{\delta(t)} \omega$ be an Abelian integral, where $H=y^2-x^{n+1}+P(x)$ is a hyperelliptic polynomial of Morse type, $\delta(t)$ a horizontal family of cycles in the curves $\{H=t\}$, and $\omega$ a polynomial 1-form in the…

We show that a criterion for an integral domain to be a principal ideal domain (PID), due to Dedekind and Hasse, can also be applied in quaternion orders, and that it can be used to build a finite algorithm to determine if a given order is…

Let $\beta\in(1,2)$ and $x\in [0,\frac{1}{\beta-1}]$. We call a sequence $(\epsilon_{i})_{i=1}^\infty\in\{0,1\}^{\mathbb{N}}$ a $\beta$-expansion for $x$ if $x=\sum_{i=1}^{\infty}\epsilon_{i}\beta^{-i}$. We call a finite sequence…

We consider integer sequences that satisfy a recursion of the form $x_{n+1} = P(x_n)$ for some polynomial $P$ of degree $d > 1$. If such a sequence tends to infinity, then it satisfies an asymptotic formula of the form $x_n \sim A…

We study generating functions of strict and non-strict order polynomials of series-parallel posets, called order series. These order series are closely related to Ehrhart series and h*-polynomials of the associated order polytopes. We…

A preferential arrangement of a set is a total ordering of the elements of that set with ties allowed. A barred preferential arrangement is one in which the tied blocks of elements are ordered not only amongst themselves but also with…

In this paper we are interested in the following problem. Let $p$ be a prime number, $S\subset \F_p$ and $\cP\subset \{P\in\F_p [X]:\deg P\le d\}$. What is the largest integer $k$ such that for all subsets $\cA, \cB$ of $\F_p$ satisfying…

In 2013 Zhi-Wei Sun conjectured that $p(n)$ is never a power of an integer when $n>1.$ We confirm this claim in many cases. We also observe that integral powers appear to repel the partition numbers. If $k>1$ and $\Delta_k(n)$ is the…