A Woodall number is a specific type of number in number theory that is defined as a number of the form \( n \cdot 2^n - 1 \), where \( n \) is a positive integer.

Wolstenholme numbers are a special sequence of natural numbers related to combinatorial mathematics and number theory. Specifically, a Wolstenholme number \(W_n\) is defined as the binomial coefficient \(\binom{2n}{n}\) for a given non-negative integer \(n\), which counts the number of ways to choose \(n\) items from a set of \(2n\) items.

The Wilson quotient is a concept in number theory related to Wilson's theorem. Wilson's theorem states that a natural number \( p \) is a prime if and only if \[ (p-1)! \equiv -1 \, (\text{mod } p). \] The Wilson quotient is computed using the factorial of \( p-1 \) and is specifically defined for prime numbers. It can be expressed as: \[ W(p) = \frac{(p-1)!

A **weird number** is a specific type of integer in number theory that has a unique property regarding its divisors. Specifically, a weird number is defined as a positive integer that is abundant, meaning that the sum of its proper divisors (factors excluding the number itself) is greater than the number, but no subset of these divisors sums to the number itself.

The Wedderburn–Etherington numbers are a sequence of integers that count certain types of binary trees, specifically the number of distinct full binary trees (or proper binary trees) with a given number of internal nodes. A full binary tree is a tree in which every internal node has exactly two children. The \( n \)-th Wedderburn–Etherington number counts the number of full binary trees with \( n \) internal nodes.

Weak ordering, in the context of preference relations and mathematics, refers to a situation in which items can be compared and ordered based on some criteria, but the order does not strictly define a comprehensive ranking. In weak ordering, two or more items can be considered equivalent in terms of preference, meaning that they can be equally preferred or ranked at the same level without establishing a definitive hierarchy among them.

The term "unusual number" can have various meanings depending on the context in which it is used, as it is not a standard mathematical term. Here are a few interpretations that could apply: 1. **Mathematical Context**: In some mathematical discussions, "unusual" might refer to numbers that exhibit unique or rare properties.

An untouchable number is a positive integer that cannot be expressed as the sum of the proper divisors (the divisors excluding the number itself) of any positive integer. In other words, there is no positive integer for which its proper divisors sum up to the untouchable number.

A **unitary perfect number** is a special type of number that is defined in relation to its divisors. To understand unitary perfect numbers, we first need to define a few terms: 1. **Unitary Divisor:** A divisor \( d \) of a number \( n \) is called a unitary divisor if \( \gcd(d, n/d) = 1 \).

Ulam numbers are a sequence of integers that start with the numbers 1 and 2. Subsequent Ulam numbers are generated using a specific rule: each Ulam number is the smallest positive integer that can be expressed as the sum of two distinct earlier Ulam numbers in exactly one way. The sequence begins as follows: 1. The first two Ulam numbers are 1 and 2.

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.

In mathematics, a "telephone number" generally refers to a method of representing numbers in a specific format that resembles a phone number. This can include various mathematical concepts, such as: 1. **Digits and Place Value**: A telephone number comprises a specific sequence of digits, often grouped into sections (like area codes, local numbers, etc.), which can be analyzed mathematically in terms of digit placement and value.

Sylvester's sequence is an integer sequence defined by a specific recursive process.

A **superperfect number** is a special type of number that is defined in number theory. It is characterized by its relationship to perfect numbers, which themselves are defined as positive integers that are equal to the sum of their proper divisors (excluding the number itself). A superperfect number is defined as a number \( n \) such that the sum of its divisors \( \sigma(n) \) (including \( n \) itself) is equal to \( 2n \).

A **superior highly composite number** is a type of positive integer that has a greater ratio of divisors to size than any smaller positive integer. In other words, a superior highly composite number has more divisors than any smaller number when the number of divisors is maximized relative to the number itself.

The term "superfactorial" is used to refer to an extension of the factorial function, similar to how tetration is an extension of exponentiation. The superfactorial of a positive integer \( n \) is denoted as \( \text{sf}(n) \) and is defined as the product of the factorials of all positive integers up to \( n \). Mathematically, it is defined as: \[ \text{sf}(n) = 1!

A **superabundant number** is a positive integer \( n \) for which the ratio of the sum of its divisors \( \sigma(n) \) to \( n \) is greater than the ratio for any smaller positive integer \( m \).

A Super-Poulet number is a special type of number that is defined in terms of prime numbers. Specifically, a Super-Poulet number is a natural number \( n \) such that \( n \) is a power of a prime \( p^k \) where \( k \geq 1 \) (i.e.

A **sum-free sequence** is a sequence of integers such that no two elements in the sequence sum to another element in the same sequence. In other words, if \( a \) and \( b \) are elements of the sequence, then \( a + b \) should not be an element of the sequence.

A Sublime number is a specific type of number in number theory that is defined as a natural number \( n \) for which the sum of its proper divisors (the divisors of \( n \) excluding \( n \) itself) is equal to \( n \) times the number of proper divisors of \( n \).