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Many natural counting problems arise in connection with the normal form of braids--and seem to have never been considered so far. Here we solve some of them by analysing the normality condition in terms of the associated permutations, their…
In this paper we continue our research on the concept of liken. This notion has been defined as a sequence of non-negative real numbers, tending to infinity and closed with respect to addition in $\mathbb{R}$. The most important examples of…
An integer $n\geq 1$ is a $v$-palindrome if it is not a multiple of $10$, nor a decimal palindrome, and such that the sum of the prime factors and corresponding exponents larger than $1$ in the prime factorization of $n$ is equal to that of…
We investigate compositions of a positive integer with a fixed number of parts, when there are several types of each natural number. These compositions produce new relationships among binomial coefficients, Catalan numbers, and numbers of…
There are several standard procedures used to create new sequences from a given sequence or from a given pair of sequences. In this paper I discuss the most popular of these procedures. For each procedure, I give a definition and provide…
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 introduce and consider a certain probability question involving elementary number theory and the likelihood that a fixed prime will appear in a certain recursively defined factorization of an integer. We derive several convergent…
By defining the dimension of natural numbers as the number of prime factors, all natural numbers smaller than 2^(n+1) (n is a natural number) can be classified by their dimensions, and the count of numbers of each dimension gives a…
Classical crystals are solid materials containing arbitrarily long periodic repetitions of a single motif. In this paper, we study the maximal possible repetition of the same motif occurring in beta-integers -- one dimensional models of…
We consider an infinite sequence of rooted trees naturally emerging in a number-theoretical context. We advance some ideas on its structure by discussing some elementary properties. Some of those properties are shown to be related to…
By means of the generating function method, a linear recurrence relation is explicitly resolved. The solution is expressed in terms of the Stirling numbers of both the first and the second kind. Two remarkable pairs of combinatorial…
A classical theorem of Kempner states that the sum of the reciprocals of positive integers with missing decimal digits converges. This result is extended to much larger families of "missing digits" sets of positive integers with both…
We discuss an unusual phenomenon in (integral) positive ternary quadratic forms. We also describe an interesting pairing of genera of ternary forms.
We search for triangular numbers that are multiples of other triangular numbers. It is found that for any positive non-square integer multiplier, there is an infinity of multiples of triangular numbers that are triangular numbers and…
We consider a family of nonlinear recurrences with the Laurent property. Although these recurrences are not generated by mutations in a cluster algebra, they fit within the broader framework of Laurent phenomenon algebras, as introduced…
In this text we study the regularity of matrices with special polynomial entries. Barring some mild conditions we show that these matrices are regular if a natural limit size is not exceeded. The proof draws connections to generalized…
We study, from the viewpoint of metrical number theory and (infinite) ergodic theory, the probabilistic laws governing the occurrence of prime numbers as digits in continued fraction expansions of real numbers.
The theory of regularity is a philosophical perspective in which laws of nature are just descriptions, that is to say, laws of nature do not govern the world. Moreover, according to the theory of regularity, the number of laws of nature…
Formal languages are sets of strings of symbols described by a set of rules specific to them. In this note, we discuss a certain class of formal languages, called regular languages, and put forward some elementary results. The properties of…
In this paper we present a use of nonstandard methods in the theory of ultrafilters and in related applications to combinatorics of numbers.