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.

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}.

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 factorial prime is a specific type of prime number that can be expressed in the form \( n! \pm 1 \), where \( n! \) represents the factorial of a non-negative integer \( n \). The two forms are \( n! - 1 \) and \( n! + 1 \). For example: - For \( n = 0 \): \( 0!

Goodstein's theorem is a result in mathematical logic and number theory that deals with a particular sequence of natural numbers known as Goodstein sequences. The theorem states that every Goodstein sequence eventually terminates at 0, despite the fact that the terms of the sequence can grow extremely large before reaching 0. To understand Goodstein's theorem, we first need to define how a Goodstein sequence is constructed: 1. **Starting Point**: Begin with a natural number \( n \).

A harmonic divisor number is a concept in number theory related to the harmonic mean of the divisors of a number. Specifically, an integer \( n \) is called a harmonic divisor number if the sum of the reciprocals of its divisors is an integer.

A **hemiperfect number** is a type of integer that is related to the concept of perfect numbers and their generalizations. Specifically, a positive integer \( n \) is considered a hemiperfect number if there exists a subset of its proper divisors (the divisors excluding itself) such that the sum of the divisors in that subset equals \( n \).

A highly composite number is a positive integer that has more divisors than any smaller positive integer. In other words, it is a number that has a greater number of divisors than all the integers less than it. The concept of highly composite numbers was introduced by the mathematician Srinivasa Ramanujan.

A home prime is a concept in number theory that relates to the representation of numbers as sums of prime numbers. Specifically, a home prime is produced by repeatedly factoring a composite number into its prime factors, then concatenating those prime factors (written in order), and repeating the process until a prime number is obtained. Here’s how it works in detail: 1. Start with a composite number. 2. Factor it into its prime factors.

Integer complexity is a concept in number theory that refers to the minimum number of ones needed to express a positive integer \( n \) using just addition, multiplication, and parentheses. The complexity of an integer is denoted as \( C(n) \). For example: - The integer \( 1 \) has a complexity of \( C(1) = 1 \) because it can be represented as simply using one "1".

The Katydid sequence, also known as the "katydid word sequence," is a specific sequence of numbers defined by a recursive process based on the number of syllables in the word "katydid." The word "katydid" has three syllables, which influences the way the sequence is constructed. To generate the Katydid sequence: 1. Start with the first term as \( a_1 = 1 \).

A Lehmer sequence is a specific type of sequence that is generated using the properties of numbers in a deterministic manner. It is defined by a recurrence relation with integer coefficients. The Lehmer sequence \( L(n) \) is typically constructed as follows: 1. The initial terms of the sequence are defined as: - \( L(0) = 0 \) - \( L(1) = 1 \) 2.

The Lucas sequence is a series of numbers that is similar to the Fibonacci sequence.

A "lucky number" is typically a number that people consider to bring good fortune or positive energy. The concept of lucky numbers varies across cultures and individuals. For example: 1. **Cultural Significance**: In some cultures, certain numbers are viewed as lucky due to traditional beliefs or superstitions. For instance, in Chinese culture, the number 8 is considered lucky because it sounds similar to the word for "prosperity.

The Ordered Bell number is a concept in combinatorial mathematics that counts the number of ways to partition a set into a certain number of non-empty ordered subsets. More formally, the \( n \)-th Ordered Bell number, denoted as \( B_n^{o} \), gives the number of ways to partition a set of size \( n \) into \( k \) non-empty subsets, where the order of the subsets matters.

A perfect totient number is a type of number related to the concept of totient functions in number theory. The totient function, denoted as \( \phi(n) \), counts the integers up to \( n \) that are coprime to \( n \).

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. In other words, a prime number is only divisible by 1 and the number itself, meaning it cannot be divided evenly by any other integers. For example, the numbers 2, 3, 5, 7, 11, and 13 are all prime numbers.

A **quasiperfect number** is a hypothetical concept in number theory. It is defined as a positive integer \( n \) for which the sum of its proper divisors (all divisors excluding the number itself) is equal to \( n + 1 \).

A sociable number is a number that forms a closed chain with other numbers through a specific process involving the sum of its proper divisors. More formally, a sociable number is part of a group of numbers where each number in the group is the sum of the proper divisors of the preceding number.

A **square-free integer** is an integer that is not divisible by the square of any prime number. In other words, a square-free integer cannot have any prime factor raised to a power greater than one in its prime factorization. For example: - The integer 30 is square-free because its prime factorization is \(2^1 \times 3^1 \times 5^1\); none of the prime factors are squared or higher.