An ergodic sequence typically refers to a sequence of random variables or a time series in the context of ergodic theory, which is a branch of mathematics and statistical mechanics. In simple terms, a sequence (or process) is said to be ergodic if, over a long period of time (or a large sample size), its time averages converge to the same value as its ensemble averages.

An Erdős–Woods number is a specific type of integer related to the study of sets of consecutive integers and their relationships to prime numbers. More formally, an integer \( n \) has an Erdős–Woods number if there exists a positive integer \( k \) such that the set of integers: \[ \{ n, n+1, n+2, \ldots, n+k \} \] contains \( k \) or more prime numbers.

The Erdős–Nicolas number is a concept from combinatorial number theory that is associated with a particular type of partitioning of the natural numbers. Specifically, it's named after the mathematicians Paul Erdős and Michel Nicolas, who studied certain properties of numbers and sequences.

An equidigital number is a positive integer \( n \) for which the number of digits in \( n \) is equal to the number of digits in its divisor structure when expressed in base 10. This concept can be further understood through the lens of its prime factorization.

An elliptic divisibility sequence (EDS) is a sequence of integers that arises from the theory of elliptic curves and has interesting divisibility properties. These sequences are generated based on the coordinates of points on an elliptic curve, typically given in Weierstrass form. The properties of EDSs are linked to the arithmetic of elliptic curves, particularly their group structure.

A doubly triangular number is a figurate number that represents a triangular pyramid. In mathematical terms, a doubly triangular number can be derived by summing triangular numbers. The \(n\)-th triangular number \(T_n\) is given by the formula: \[ T_n = \frac{n(n + 1)}{2} \] Doubly triangular numbers can also be expressed in a closed formula.

The double factorial, denoted by \( n!! \), is a mathematical operation that is defined for non-negative integers. It is the product of all the integers from \( n \) down to 1 that have the same parity (odd or even) as \( n \). Specifically, it is defined as follows: 1. For an even integer \( n = 2k \): \[ n!! = 2k!!

A Double Mersenne number is a special class of numbers that is defined using Mersenne numbers. Mersenne numbers are of the form \( M_n = 2^n - 1 \), where \( n \) is a positive integer. A Double Mersenne number is then defined as a Mersenne number whose index \( n \) itself is a Mersenne prime.

A **divisibility sequence** is a sequence of integers \( (a_n) \) where each term divides the subsequent terms in the sequence according to specific criteria. More formally, a sequence \( (a_n) \) is called a divisibility sequence if for each pair of indices \( m < n \), the term \( a_m \) divides \( a_n \) (denoted as \( a_m \mid a_n \)).

Dirichlet series inversion is a method in analytic number theory that relates Dirichlet series and arithmetic functions. A Dirichlet series is a series of the form \[ F(s) = \sum_{n=1}^\infty \frac{a(n)}{n^s} \] where \( a(n) \) represents a sequence of complex numbers and \( s \) is a complex variable.

A Descartes number is a particular type of geometric configuration related to the curvature of circles. The concept arises from the Cartesian circle theorem, and it specifically pertains to a set of circles that are tangent to each other.

A derangement is a specific type of permutation of a set of elements where none of the elements appear in their original position. In other words, if you have a set of objects and wish to rearrange them such that no object remains in its initial position, that arrangement is referred to as a derangement. For example, consider the set of objects {1, 2, 3}.

Delannoy numbers are a type of combinatorial number that counts the number of different paths from the bottom-left corner to the top-right corner of an \( m \times n \) grid, where you can move only to the right, up, or diagonally up-right at each step. The Delannoy number \( D(m, n) \) represents the total number of such paths.

A deficient number is a positive integer \( n \) for which the sum of its proper divisors (excluding itself) is less than \( n \).

A Dedekind number, denoted as \(M(n)\), is a specific type of combinatorial object that counts the number of ways to partition the power set of an \(n\)-element set into antichains, which are sets of subsets where no one subset is contained within another.

Cullen numbers are a sequence of integers that are defined by the formula: \[ C_n = n \cdot 2^n + 1 \] where \( n \) is a non-negative integer (i.e., \( n = 0, 1, 2, 3, \ldots \)).

A coordination sequence is a term most commonly used in the context of mathematical structures such as graphs, networks, or crystal lattices. It describes the number of nearest neighbors (or connected vertices) that a particular vertex has at various levels of distance from it.

Congruum could refer to a few different concepts depending on the context. In mathematical terms, it can refer to congruence, which is a relation that indicates that two numbers or shapes are equivalent in some sense, often in terms of size or shape. In geometry, for example, two triangles are said to be congruent if they have the same shape and size, regardless of their position or orientation.

The term "complete sequence" can refer to various concepts depending on the context in which it is used. Here are a few possible interpretations: 1. **Mathematics**: In mathematics, a complete sequence might refer to a series of numbers or functions that are fully specified or encompass all necessary elements within a particular set. For example, in the context of sequences, a complete sequence of integers would include every integer within a specified range.

A colossally abundant number is a special type of integer that surpasses a specific threshold related to its divisors. More formally, a positive integer \( n \) is considered colossally abundant if it satisfies the condition: \[ \frac{\sigma(n)}{n} > \frac{\sigma(m)}{m} \] for all positive integers \( m < n \), where \( \sigma(n) \) is the sum of the positive divisors of \( n \).