A centered octagonal number is a type of figurate number that represents a pattern of dots arranged in an octagonal shape. The formula to find the nth centered octagonal number is given by: \[ C_n = 3n^2 - 3n + 1 \] where \(C_n\) is the nth centered octagonal number and \(n\) is a positive integer (1, 2, 3, ...).

A centered triangular number is a specific type of figurate number that represents a triangular figure with a center point. Centered triangular numbers are generated by arranging dots in the shape of a triangle with a single dot in the center and additional layers of dots forming outer triangular frames.

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

Statistical indicators are quantitative measures that provide insights into various aspects of data and help in analyzing patterns, trends, and relationships within that data. They are often used in research, economics, social sciences, healthcare, and many other fields to summarize information, facilitate decision-making, and assess performance. Here are some key characteristics and types of statistical indicators: ### Characteristics: 1. **Quantitative**: They provide numerical data that can be analyzed statistically.

The Education Index is a composite measure used to assess the level of educational attainment and the quality of education in a particular region or country. It is part of the Human Development Index (HDI) and serves to provide insights into the overall development and well-being of a population. The Education Index typically comprises two key indicators: 1. **Mean Years of Schooling**: This measures the average number of years of education received by people aged 25 and older in a given population.

The G-index is a metric used to assess the productivity and citation impact of academic publications. It is an enhancement of the more commonly known h-index. The G-index was proposed by Leo Egghe in 2006 and aims to address some of the limitations of the h-index. ### Definition: The G-index is defined such that a researcher has a G-index of "g" if they have published "g" papers that have each received, on average, at least "g" citations.

The Renkonen similarity index is a measure used to quantify the similarity between two or more samples based on the presence and abundance of species or other categorical data. It was developed in the context of ecological studies to compare community compositions.

An Achilles number is a positive integer that is a powerful number but not a perfect power. A powerful number is defined as a number \( n \) such that in its prime factorization, every prime number \( p \) appears with an exponent of at least 2. In contrast, a perfect power is a number of the form \( m^k \) where \( m \) and \( k \) are positive integers and \( k \geq 2 \).

The Beatty sequence is a sequence of numbers that can be derived from the mathematical concept of filling the real line with two sequences whose terms are the floor functions of the multiples of two irrational numbers.

Betrothed numbers are a pair of positive integers \( (m, n) \) such that each number plus one equals the sum of the other number's proper divisors. In formal terms, if \( \sigma(n) \) denotes the sum of the divisors of \( n \), then \( m \) and \( n \) are betrothed if the following conditions hold: 1. \( \sigma(m) - m = n + 1 \) 2.

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 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 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.

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.

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.

The number 104 is an integer that comes after 103 and before 105. It is an even number and can be factored into its prime components as \(2^3 \times 13\).

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 \( n + 1 \).