Related papers: On Integer Sequences Associated To Two Distinct Su…

Permutations are usually enumerated by size, but new results can be found by enumerating them by inversions instead, in which case one must restrict one's attention to indecomposable permutations. In the style of the seminal paper by Simion…

We give a new short proof of the most simple relation between consecutive power sums of the first m positive integers.

To partition a sequence of n integers into subsets with prescribed sums is an NP-hard problem in general. In this paper we present an efficient solution for the homogeneous version of this problem; i.e. where the elements in each subset add…

All integer solutions $\left(M,a,c\right)$ to the problem of the sums of $M$ consecutive cubed integers $\left(a+i\right)^{3}$ ($a>1$, $0\leq i\leq M-1$) equaling squared integers $c^{2}$ are found by decomposing the product of the…

Let $\mathcal{A}=\{a_{n}\}_{n=1}^{\infty}$ and $\mathcal{B}=\{b_{n}\}_{n=1}^{\infty}$ be two sequences of positive integers (not necessarily distinct). Under some restrictions on $\mathcal{A}$ and $\mathcal{B}$, we obtain a lower bound for…

We give a new characterization of generalized B\"uchi sequences (sequences whose sequence of squares has constant second difference $(a)$, for some fixed integer $a$) of length 3 over the integers and a strategy for attacking B\"uchi's n…

Let q>1 and m>0 be relatively prime integers. We find an explicit period $\nu_m(q)$ such that for any integers n>0 and r we have $[n+\nu_m(q),r]_m(a)=[n,r]_m(a) (mod q)$ whenever a is an integer with $\gcd(1-(-a)^m,q)=1$, or a=-1 (mod q),…

Which integers can be written as a quotient of sums of distinct powers of three? We outline our first steps toward an answer to this question, beginning with a necessary and almost sufficient condition. Then we discuss an algorithm that…

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…

Partitioning a set into similar, if not, identical, parts is a fundamental research topic in combinatorics. The question of partitioning the integers in various ways has been considered throughout history. Given a set $\{x_1, \ldots, x_n\}$…

One of the most basic results in graph theory states that every graph with at least two vertices has two vertices with the same degree. Since there are graphs without $3$ vertices of the same degree, it is natural to ask if for any fixed…

In this paper we study the equation $$ x^k + (x+1)^k = y^n,\quad n\geq 3, $$ when $k\equiv 2\pmod{4}$. We prove that the only solutions are for $x=0, -1$ when $6\leq k\leq 100$ or for a $k$ with odd prime factors congruent to $3\pmod{4}$.…

This paper is concerned with the existence of consecutive pairs and consecutive triples of multiplicatively dependent integers. A theorem by LeVeque on Pillai's equation implies that the only consecutive pairs of multiplicatively dependent…

A new polynomial sieve is presented and used to show that almost all integers have at most one representation as a sum of two values of a given polynomial of degree at least 3.

The sequence of 1/2-discrepancy sums of $\{x + i \theta \bmod 1\}$ is realized through a sequence of substitutions on an alphabet of three symbols; particular attention is paid to $x=0$. The first application is to show that any asymptotic…

In this article, constant dimension subspace codes whose codewords have subspace distance in a prescribed set of integers, are considered. The easiest example of such an object is a {\it junta}; i.e. a subspace code in which all codewords…

Although 10^230 terms of Recaman's sequence have been computed, it remains a mystery. Here three distant cousins of that sequence are described, one of which is also mysterious. (i) {A(n), n >= 3} is defined as follows. Start with n, and…

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 $\lambda > 2$…

We extend results of Jagy and Kaplansky and the present authors and show that for all $k\geq 3$ there are infinitely many positive integers $n$, which cannot be written as $x^2+y^2+z^k=n$ for positive integers $x,y,z$, where for…

In this article we consider questions related to the behavior of the moments $M_{m}\left( \left\{ z_{j}\right\} \right) $ when the indices are restricted to specific subsequences of integers, such as the even or odd moments. If $n\geq2$ we…