Related papers: Perfect Skolem sets

This paper provides a complete solution to Skolem's problem for the $k$-generalized Lucas sequence $(L_n^{(k)})_{n \in \mathbb{Z}}$ with a primary focus on its behavior at negative indices. We characterize the zero-distribution of this…

Fix a positive real number $\theta$. The natural numbers $m$ with largest square-free divisor not exceeding $m^\theta$ form a set $\mathscr{A}$, say. It is shown that whenever $\theta>1/2$ then all large natural numbers $n$ are the sum of…

Let S=K[x_1,...,x_n] be a polynomial ring. Denote by $p_a$ the power sum symmetric polynomial x_1^a+...+x_n^a. We consider the following two questions: Describe the subsets $A \subset \mathbb{N}$ such that the set of polynomials $p_a$ with…

For almost a century, the decidability of the Skolem Problem - that is, the problem of finding whether a given linear recurrence sequence (LRS) has a zero term - has remained open. A breakthrough in the 1980s established that the Skolem…

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…

The Ulam sequence is given by $a_1 =1, a_2 = 2$, and then, for $n \geq 3$, the element $a_n$ is defined as the smallest integer that can be written as the sum of two distinct earlier elements in a unique way. This gives the sequence $1, 2,…

For a fixed integer $k$, we define a sequence $A_k=(a_k(n))_{n\geq0}$ and a corresponding sparse subsequence $S_k$ using the cardinality of the $n$-th symmetric power of the set $\{1,2,\ldots, k\}$. For $k\in\{2,\dots,8\}$, we find…

A sequence $(a_1, \ldots, a_n)$ of nonnegative integers is an {\em ascent sequence} if $a_0 =0$ and for all $i \geq 2$, $a_i$ is at most 1 plus the number of ascents in $(a_1, \ldots, a_{i-1})$. Ascent sequences were introduced by…

Golomb's sequence is the unique nondecreasing sequence of positive integers in which each $n$ appears exactly $a(n)$ times. It satisfies the global self-referential rule \[ a\bigl(a(n)+a(n-1)+\cdots+a(1)\bigr)=n, \] grows smoothly like a…

In this paper, given a simple linear recurrence sequence of algebraic numbers, which has either a dominant characteristic root or exactly two characteristic roots of maximal modulus, we give some explicit lower bounds for the index beyond…

The Stern sequence (s(n)) is defined by s(0) = 0, s(1) = 1, s(2n) = s(n), s(2n+1) = s(n) + s(n+1). Stern showed in 1858 that gcd(s(n),s(n+1)) = 1, and that for every pair of relatively prime positive integers (a,b), there exists a unique n…

A set A of positive integers is called a perfect difference set if every nonzero integer has an unique representation as the difference of two elements of A. We construct dense perfect difference sets from dense Sidon sets. As a consequence…

The Skolem Problem asks to determine whether a given linear recurrence sequence (LRS) $\langle u_n \rangle_{n=0}^\infty$ over the integers has a zero term, that is, whether there exists $n$ such that $u_n = 0$. Decidability of the problem…

Consider the recursive relation generating a new positive integer $n_{\ell +1}$ from the positive integer $n_{\ell }$ according to the following simple rules: if the integer $n_{\ell }$ is odd, $n_{\ell +1}=3n_{\ell }+1$; if the integer…

We isolate conditions on the relative size of sets of natural numbers $A,B$ that guarantee a nonempty intersection $\Delta(A)\cap\Delta(B)\ne\emptyset$ of the corresponding sets of distances. Such conditions apply to a large class of zero…

This short note deals with the so-called $ Sock \; Matching \; Problem$. We define $B_{n,k}$ as the number of all the finite sequences $a_1, \ldots, a_{2n}$ of nonnegative integers which contain at least one occurrence of $k$ $(1 \leq k…

The study of perfect numbers (numbers which equal the sum of their proper divisors) goes back to antiquity, and is responsible for some of the oldest and most popular conjectures in number theory. We investigate a generalization introduced…

A Universal Cycle for t-multisets of [n]={1,...,n} is a cyclic sequence of $\binom{n+t-1}{t}$ integers from [n] with the property that each t-multiset of [n] appears exactly once consecutively in the sequence. For such a sequence to exist…

Let $L=(L_d)_{d \in \mathbb N}$ be any ordered probability sequence, i.e., satisfying $0 < L_{d+1} \le L_d$ for each $d \in \mathbb N$ and $\sum_{d \in \mathbb N} L_d =1$. We construct sequences $A = (a_i)_{i \in \mathbb N}$ on the…

Suppose we are given two probability measures on the set of one-way infinite finite-alphabet sequences and consider the question when one of the measures predicts the other, that is, when conditional probabilities converge (in a certain…