Related papers: A Hidden Signal in the Ulam sequence

Let $m\in\mathbb{Z}$ be an integer and $L_m=\mathbb{Q}(\alpha)$ be the simplest cubic field with class number $h_m$ and conductor $\mathfrak{f}_m$ where $\alpha$ is a root of $f_m(X)=X^3-mX^2-(m+3)X-1$. Let $\mathcal{O}_{L_m}$ be the ring…

Given a recurrent sequence ${\bf U}:=\{U_n\}_{n\ge 0}$ we consider the problem of counting ${\mathcal M}_U(x)$, the number of integers $n\le x$ such that $U_n=u^2+nv^2$ for some integers $u,v$. We will show that ${\mathcal M}_U(x)\ll x(\log…

The Fibonacci cube of dimension n, denoted as $\Gamma\_n$, is the subgraph of the hypercube induced by vertices with no consecutive 1's. The irregularity of a graph G is the sum of |d(x)-d(y)| over all edges {x,y} of G. In two recent paper…

Ulam asked for the maximum number of questions required to determine an integer between one and one million by asking questions whose answer is `Yes' or `No' and where one untruthful answer is allowed. Pelc showed that the number of…

A Sidon set is a set A of integers such that no integer has two essentially distinct representations as the sum of two elements of A. More generally, for every positive integer g, a B_2[g]-set is a set A of integers such that no integer has…

In this paper we determine the number and typical structure of sets of integers with bounded doubling. In particular, improving recent results of Green and Morris, and of Mazur, we show that the following holds for every fixed $\lambda > 2$…

The size $b$ of the smallest bidirectional macro scheme, which is arguably the most general copy-paste scheme to generate a given sequence, is considered to be the strictest reachable measure of repetitiveness. It is strictly lower-bounded…

The order $O_n(\sigma)$ of a permutation $\sigma$ of $n$ objects is the smallest integer $k \geq 1$ such that the $k$-th iterate of $\sigma$ gives the identity. A remarkable result about the order of a uniformly chosen permutation is due to…

A covering system of the integers is a finite collection of modular residue classes $\{a_m \bmod{m}\}_{m \in S}$ whose union is all integers. Given a finite set $S$ of moduli, it is often difficult to tell whether there is a choice of…

For a finite (not necessarily Abelian) group $(\Gamma,\cdot)$, let $n(\Gamma) \in \mathbb{N}$ denote the smallest positive integer $n$ such that for every labelling of the arcs of the complete digraph of order $n$ using elements from…

The normality measure $\mathcal{N}$ has been introduced by Mauduit and S{\'a}rk{\"o}zy in order to describe the pseudorandomness properties of finite binary sequences. Alon, Kohayakawa, Mauduit, Moreira and R{\"o}dl proved that the minimal…

The $n$th Ramanujan prime is the smallest positive integer $R_n$ such that if $x \ge R_n$, then there are at least $n$ primes in the interval $(x/2,x]$. For example, Bertrand's postulate is $R_1 = 2$. Ramanujan proved that $R_n$ exists and…

In $1963$ Graham proved that every positive integer $n \ge 78$ can be written as a sum of distinct positive integers $a_1, a_2, \ldots, a_r$ for which $\frac{1}{a_1} + \frac{1}{a_2} + \ldots + \frac{1}{a_r}$ is equal to $1$. In the same…

A Sidon sequence is a sequence of integers a_1 < a_2 < a_3 < ... with the property that the sums a_i+a_j (i\le j) are distinct. This work contains a survey of Sidon sequences and their generalizations, and an extensive annotated and…

Define $\|n\|$ to be the complexity of $n$, the smallest number of ones needed to write $n$ using an arbitrary combination of addition and multiplication. John Selfridge showed that $\|n\|\ge 3\log_3 n$ for all $n$, leading this author and…

For a weight-set $A\subseteq \mathbb Z_n$, the $A$-weighted zero-sum constant $C_A(n)$ is defined to be the smallest natural number $k$, such that any sequence of $k$ elements in $\mathbb Z_n$ has an $A$-weighted zero-sum subsequence of…

We study here the so called subsequence pattern matching also known as hidden pattern matching in which one searches for a given pattern $w$ of length $m$ as a subsequence in a random text of length $n$. The quantity of interest is the…

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…

For a set of positive integers $D$, a $k$-term $D$-diffsequence is a sequence of positive integers $a_1<a_2<\cdots<a_k$ such that $a_i-a_{i-1}\in D$ for $i=2,3,\cdots,k$. For $k\in\mathbb{Z}^+$ and $D\subset \mathbb{Z}^+$, we define…

The (Mahler-Popken) complexity $\| n \|$ of a natural number $n$ is the smallest number of ones that can be used via combinations of multiplication and addition to express $n$, with parentheses arranged in such a way so as to form legal…