The term "Fermi-Dirac prime" refers to a specific type of prime number that arises from the Fermi-Dirac distribution, which is a statistical distribution that describes the occupancy of energy levels by fermions (particles that follow the Pauli exclusion principle, such as electrons). In more detail, the Fermi-Dirac distribution is used in quantum statistics to describe how particles occupy quantum states at thermal equilibrium, especially at absolute zero temperature.

A Fortunate number is a concept from number theory that refers to a positive integer \( n \) such that \( n + 1 \) is either a prime number or is a prime power (a number of the form \( p^k \) where \( p \) is a prime and \( k \) is a positive integer). Essentially, the Fortunate numbers are obtained by adding 1 to the numbers in the sequence of primes or prime powers.

A Genocchi number is a particular type of integer that arises in number theory and is related to the Bernoulli numbers. Specifically, the Genocchi numbers \(G_n\) are defined as the integers that can be expressed through the generating function: \[ \frac{2x}{e^x + 1} = \sum_{n=0}^{\infty} G_n \frac{x^n}{n!

A highly cototient number is a natural number \( n \) such that the equation \( x - \varphi(x) = n \) has more solutions than any smaller positive integer \( m \). Here, \( \varphi(x) \) is the Euler's totient function, which counts the number of integers up to \( x \) that are relatively prime to \( x \).

The Hofstadter sequence is a family of sequences named after the American computer scientist Douglas Hofstadter, who introduced it in his book "Gödel, Escher, Bach: An Eternal Golden Braid." There are several variations of Hofstadter sequences, but one of the most well-known is the Hofstadter Q-sequence, defined recursively as follows: 1. \( Q(1) = 1 \) 2. \( Q(2) = 1 \) 3.

An integer sequence is a list of numbers arranged in a specific order, where each number in the list (called a term) is an integer. Integer sequences can be defined in various ways, such as by a formula, a recurrence relation, or by specifying initial terms.

A **K-regular sequence** is a specific type of sequence defined in the context of combinatorial number theory and formal language theory. More formally, a sequence is said to be K-regular if it can be recognized by a finite automaton or if it satisfies certain algebraic properties that can be expressed using K-dimensional vectors or matrices. The most common definition of K-regular sequences comes from the context of **generating functions**.

A list of integer sequences typically refers to various collections of sequences made up of integers that follow specific patterns or rules. These sequences can be found in mathematical literature and often have interesting properties or applications in number theory, combinatorics, and computer science. One prominent source for integer sequences is the **OEIS (Online Encyclopedia of Integer Sequences)**, which catalogs a vast number of integer sequences along with their definitions, formulas, theorems, and historical context.

Lucky numbers are a sequence of natural numbers that are generated by a specific sieve process, first introduced by the mathematician Leonhard Euler. The process of generating lucky numbers is similar to that used in the Sieve of Eratosthenes for finding prime numbers, but instead of eliminating multiples of prime numbers, it eliminates numbers based on their positions.

The Mian–Chowla sequence is an infinite sequence of integers defined by a specific recursive relationship. The sequence is constructed in such a way that it avoids repetitions and maintains specific properties regarding sums of elements. The definition of the Mian–Chowla sequence can be outlined as follows: 1. The first element of the sequence is 1, i.e., \( a_1 = 1 \).

The Narayana numbers are a sequence of numbers that appear in combinatorial mathematics and are related to various counting problems, including those involving paths and combinations.

A **nontotient** is a positive integer \( n \) for which there is no integer \( k \) such that \( k \) and \( n \) are coprime, and \( \phi(k) = n \), where \( \phi \) is the Euler's totient function. The Euler's totient function \( \phi(k) \) counts the number of integers up to \( k \) that are coprime to \( k \).

The term "power of 10" refers to expressions that represent numbers in the form of \(10^n\), where \(n\) is an integer. The power indicates how many times the base (10) is multiplied by itself.

A primary pseudoperfect number is a type of integer closely related to the concepts of number theory, particularly with respect to the properties of its divisors. A positive integer \( n \) is called a primary pseudoperfect number if it can be expressed as the sum of a subset of its proper divisors (the divisors excluding itself) plus one.

"Singly even" and "doubly even" typically refer to types of numbers in the context of mathematics, particularly in discussing properties of integers or sets of integers. 1. **Singly Even Numbers**: A number is termed "singly even" if it is divisible by 2 but not by 4. In other words, singly even numbers can be expressed in the form \(4k + 2\), where \(k\) is an integer.

A sparsely totient number is a positive integer \( n \) for which the ratio of the Euler's totient function \( \varphi(n) \) to \( n \) is relatively small compared to other integers. More formally, a number \( n \) is considered a sparsely totient number if: \[ \frac{\varphi(n)}{n} < \frac{1}{\log n} \] for sufficiently large \( n \).

A Thabit number is a specific type of integer that is part of a mathematical sequence defined by certain properties. The Thabit numbers are related to the Fibonacci sequence, specifically by being represented as a summation involving Fibonacci numbers. Formally, the n-th Thabit number \( T_n \) can be defined as: \[ T_n = \sum_{k=1}^{n} F_k \] where \( F_k \) denotes the k-th Fibonacci number.

Zero (0) is a number that represents a null quantity or the absence of value. It serves several important roles in mathematics and various number systems. Here are some key aspects of zero: 1. **Identity Element**: In addition, zero is the additive identity, meaning that when you add zero to any number, the value of that number remains unchanged (e.g., \(x + 0 = x\)).

The number 1 is a basic numerical value that represents a single unit or a single whole. It is the first positive integer and is often used as a foundational element in mathematics. In various contexts, 1 can denote unity, identity, or singularity. For example: - In arithmetic, it is the multiplicative identity, meaning any number multiplied by 1 remains unchanged. - In set theory, a set with one element has a cardinality of 1.

A quadratic integer is a type of algebraic integer that is a root of a monic polynomial of degree two with integer coefficients. In simpler terms, a quadratic integer can be expressed in the form \( a + b\sqrt{d} \), where \( a \) and \( b \) are integers, and \( d \) is a square-free integer (i.e., \( d \) is not divisible by the square of any prime).