4 citations · 16 across the 19 of their papers we have counts for
19 papers
Calculating the natural density of Mersenne numbers using nonstandard mathematical analysis
Gennady Eremin
Currently, among the open (unsolved) problems in number theory is the following: it is unknown what is the natural density of the sequence of Mersenne numbers in the set of natural…
Dyck Numbers, IV. Nested patterns in OEIS A036991
Gennady Eremin
Dyck paths (also balanced brackets and Dyck words) are among the most heavily studied Catalan families. This paper is a continuation of [2, 3, 4]. In the paper we are dealing with…
Dyck Numbers, III. Enumeration and bijection with symmetric Dyck paths
Gennady Eremin
Dyck paths (also balanced brackets and Dyck words) are among the most heavily studied Catalan families. This paper is a continuation of [2, 3]. In the paper we enumerate the terms…
Dyck Numbers, II. Triplets and Rooted Trees in OEIS A036991
Gennady Eremin
Dyck paths are among the most heavily studied Catalan families. This paper is a continuation of [2]. In the paper we are dealing with the numbering of Dyck paths, the terms of the…
Dyck Numbers, I. Successor Function
Gennady Eremin
Dyck paths are among the most heavily studied Catalan families. In the paper we are dealing with the minimal numbering of Dyck paths, with the resulting numbers, the terms of the O…
Dyck path factorization
Gennady Eremin
Dyck paths are among the most heavily studied Catalan families. We work with peaks and valleys to uniquely decompose Dyck paths into the simplest objects - prime fragments with a s…
Walking in the OEIS: From Motzkin numbers to Fibonacci numbers. The "shadows" of Motzkin numbers
Gennady Eremin
In this paper, we consider nine OEIS sequences, the analysis of which allows us to find a connection between Motzkin numbers and Fibonacci numbers. In each Motzkin number, we disti…
Arithmetization of well-formed parenthesis strings. Motzkin Numbers of the Second Kind
Gennady Eremin
In this paper, we perform an arithmetization of well-formed parenthesis strings with zeros (Motzkin words) and of corresponding Motzkin paths. The transformations used are reminisc…
4D Dyck triangle and its projections
Gennady Eremin
The classic Dyck triangle, the Catalan triangle, and the Catalan convolution matrix are plane projections of the multidimensional Dyck triangle. In the Dyck path, each node is uniq…
Decomposition of the Catalan number into the sum of squares
Gennady Eremin
Analysis of the dynamics of the Dyck words helped solve the problem of representing the Catalan number as a sum of squares of natural numbers. In this case, the Dyck triangle is co…
Naturalized bracket row and Motzkin triangle
Gennady Eremin
The paper considers the sequence of the Motzkin words, which is constructed according to formal features of natural numbers. We investigate the decomposition of well-formed parenth…
Factoring Middle Binomial Coefficients
Gennady Eremin
The article describes prime intervals into the prime factorization of the middle binomial coefficient. Prime factors and prime powers are distributed in layers. Each layer consists…
Generating function for Naturalized Series: The case of Ordered Motzkin Words
Gennady Eremin
We continue to consider the ordered lexicographic sequence, which is constructed according to the formal characteristics of a series of natural numbers. For analysis, we selected b…
Mathematics of Balanced Parentheses: The case of Ordered Motzkin Words
Gennady Eremin
It is known that the core of mathematics is natural numbers. And everything related to the natural number is interesting to mathematicians. In this paper, we draw parallels between…
Arithmetic on Balanced Parentheses: The case of Ordered Motzkin Words
Gennady Eremin
We establish a total lexicographical order on the set of Motzkin words. Elements are ordered similarly to Natural Numbers in accordance with known rules (axioms). As a result, we w…
Dynamics of balanced parentheses, lexicographic series and Dyck polynomials
Gennady Eremin
The article deals with a lexicographic order in various sequences. Consider the axiomatic of lexicographic series, based on the properties of the natural numbers. Elements of the s…
Factoring Catalan numbers
Gennady Eremin
The paper describes a prime factorization of the Catalan numbers. Odd prime factors are distributed in layers in accordance with Legendre's formula. The content of each layer is a…
Legendre's formula and -adic analysis
Gennady Eremin
In number theory, we know Legendre's formula , which calculates the -adic valuation of the factorial, i.e. the exponent…
Dyck Paths in Four-Dimensional Space
Gennady Eremin
In analyzing balanced parentheses, we consider a group of related variables in Dyck paths. In the four-dimensional space, the Dyck triangle is constructed, i.e. an integer lattice…