Related papers: A Set of Sequences in Number Theory
We prove that several results in different areas of number theory such as the divergent series, summation of arithmetic functions, uniform distribution modulo one and summation over prime numbers which are currently considered to be…
In this work initial numbers and repunit numbers have been studied. All numbers have been considered in a decimal notation. The problem of simplicity of initial numbers has been studied. Interesting properties of numbers repunit are proved:…
This article gives a brief introduction to the On-Line Encyclopedia of Integer Sequences (or OEIS). The OEIS is a database of nearly 90,000 sequences of integers, arranged lexicographically. The entry for a sequence lists the initial terms…
In this paper we study a sequence involving the prime numbers by deriving two asymptotic formulas and finding new upper and lower bounds, which improve the currently known estimates.
An inversion sequence of length $n$ is an integer sequence $(a_1, \ldots, a_n)$ such that $0 \le a_i < i$ for all $i$. The study of pattern-avoiding inversion sequences was initiated in 2015 by Mansour and Shattuck and in 2016 by Corteel,…
In this paper, we study sequences of positive numbers preserving summability. In particular, the open set property for such a family of sequences is shown. Several classes of sequences preserving summability, including polynomials, sums of…
Denote by $\mathbb{N}$ and $\mathbb{P}$ the set of all positive integers and prime numbers, respectively. Let $\mathbb{P}=\{p_1<p_2<\dots <p_n<\dots\}$, where $p_n$ is the $n$-th prime number. For $k\in\mathbb{N}$ we recursively define…
Sequences have become first class citizens in supervised learning thanks to the resurgence of recurrent neural networks. Many complex tasks that require mapping from or to a sequence of observations can now be formulated with the…
Throughout history, recreational mathematics has always played a prominent role in advancing research. Following in this tradition, in this paper we extend some recent work with crazy sequential representations of numbers- equations made of…
We provide a formal introduction into the classic theorems of general topology and its axiomatic foundations in set theory. Starting from ZFC, the exposition in this first part includes relation and order theory as well as a construction of…
Infinite sequences are of tremendous theoretical and practical importance, and in the Information Age sequences of 0s and 1s are of particular interest. Over the past century, the field of symbolic dynamics has developed to study sequences…
A graph $G$ is defined encapsulating the number theoretic notion of the Fundamental Theorem of Arithmetic. We then provide a graph theoretic approach to the fundamental results on the coprimality of two natural numbers, through the use of…
A general theory of programs, programming and programming languages built up from a few concepts of elementary set theory. Derives, as theorems, properties treated as axioms by classic approaches to programming. Covers sequential and…
The chronicle of prime numbers travel back thousands of years in human history. Not only the traits of prime numbers have surprised people, but also all those endeavors made for ages to find a pattern in the appearance of prime numbers has…
We develop new aspects of the the of numerosity theory; more exactly, we emphasize its relation with the ordinal numbers, cardinal numbers, hyperreal numbers and surreal numbers. In particular, we combine the notion of numerosity with the…
We survey current developments in the approximation theory of sequence modelling in machine learning. Particular emphasis is placed on classifying existing results for various model architectures through the lens of classical approximation…
We develop a new, powerful method for counting elements in a multiset. As a first application, we use this algorithm to study the number of occurrences of patterns in a permutation. For patterns of length 3 there are two Wilf classes, and…
This book is organized into seven chapters. Chapter one is introductory in content. The notion of neutrosophic set linear algebras and neutrosophic neutrosophic set linear algebras are introduced and their properties analysed in chapter…
First we define a new kind of function over $\mathbb{N}$. For each $i\in\mathbb{N}$ we have an associated function, which will be called $S_i$ . Then we define a new kind of sequence, to be made from the functions $S_i$ . Finally, we will…
In additive number theory, a finite set $A$ of integers is an $h$-basis for $n$ if every integer in $\{0,1,2,\ldots, n\}$ can be represented as the sum of exactly $h$ not necessarily distinct elements of $A$. This paper introduces a new…