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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.

In number theory, the prime omega function, denoted as \(\omega(n)\), counts the number of distinct prime factors of a positive integer \(n\). For example: - \(\omega(12) = 2\) because the prime factorization of 12 is \(2^2 \times 3^1\), which has the distinct prime factors 2 and 3.

A **prime power** is a number that can be expressed in the form \( p^k \), where \( p \) is a prime number and \( k \) is a positive integer. In other words, a prime power is a number that results from raising a prime number to an integer exponent greater than zero.

A Primefree sequence, also known as a prime-free sequence, is a sequence of natural numbers that does not contain any prime numbers. In other words, every number in a primefree sequence is either 1 or a composite number. The concept of primefree sequences is often used in number theory and can serve various applications, such as studying properties of composite numbers or analyzing growth rates of integer sequences without primes.

A *primitive abundant number* is a specific type of integer that has a certain relationship to its divisors. An integer \( n \) is termed an abundant number if the sum of its proper divisors (the divisors of \( n \) excluding \( n \) itself) is greater than \( n \).

A **primitive permutation group** is a specific type of group in abstract algebra, particularly within the field of group theory. A permutation group acts on a set, which is usually a set of points, and is said to be primitive if it satisfies certain conditions concerning the ways in which it partitions the set. More formally, a permutation group \( G \) acting on a set \( X \) is called **primitive** if it preserves the structure of the set in a fundamental way.

A primorial is a product of the first \( n \) prime numbers. It is denoted as \( p_n\# \), where \( p_n \) is the \( n \)-th prime number.

A **primorial prime** is a type of prime number that can be expressed in the form \( p_n\# + 1 \) or \( p_n\# - 1 \), where \( p_n \# \) (the primorial of \( p_n \)) is the product of the first \( n \) 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 \( n + 1 \).

Recamán's sequence is a well-known mathematical sequence defined recursively. It is named after the mathematician Bernardo Recamán. The sequence is defined as follows: 1. The first term \( a(0) \) is 0: \( a(0) = 0 \) 2.

A **refactorable number** is a positive integer \( n \) such that \( n \) can be divided by the number of its divisors. In mathematical terms, if \( d(n) \) denotes the number of divisors of \( n \), then \( n \) is refactorable if \( n \) is divisible by \( d(n) \) (i.e., \( n \mod d(n) = 0 \)).

A "rough number" typically refers to an estimate or an approximation that is not exact. It is often used in various contexts where precision is not crucial, and a general idea or ballpark figure suffices. For example, in financial discussions, one might provide a rough number when discussing budget estimates, costs, or statistical data, indicating that the figures are intended to give a sense of scale rather than a precise measurement.

The Schröder numbers are a sequence of numbers in combinatorial mathematics that count certain types of lattice paths or combinatorial structures. Specifically, they can be used to count the number of ways to connect points in a grid using non-crossing paths that adhere to specific restrictions.

The Schröder–Hipparchus number, denoted \( \text{SH}(n) \), is a sequence of numbers that counts the different ways to draw non-crossing partitions of a set with \( n \) elements. Specifically, these numbers are related to various combinatorial structures, including certain types of trees and the enumeration of non-crossing partitions.

A semiperfect number, also known as a weakly perfect number, is a type of integer that can be defined in the context of its divisors. Specifically, a positive integer \( n \) is considered semiperfect if the sum of some of its divisors (excluding the number itself) is equal to \( n \). For example, consider the number 12.

"Sequences" is a book written by American author and poet, John R. McTavish. It comprises a collection of poems that explore various themes, including nature, humanity, and the interconnectedness of life. The work delves into the experiences and emotions that shape human existence, often employing vivid imagery and reflective language.

"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 **smooth number** (or **friable number**) is a positive integer that has no prime factors larger than a certain number. In mathematical terms, a number \( n \) is called \( B \)-smooth if all of its prime factors are less than or equal to \( B \).

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.

The Somos sequence refers to a family of recursively defined sequences discovered by the mathematician Edward Somos. They are notable for their interesting properties and connections to combinatorial mathematics and number theory.