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.

Patchy particles are a type of colloidal particle or nanostructure that possess specific, localized regions or "patches" with distinct chemical or physical properties. These patches can be designed to have different functionalities—such as hydrophobic or hydrophilic characteristics, or specific binding affinities—for the purpose of creating complex structures or assemblies. The unique surface properties of patchy particles allow them to interact selectively with other particles or molecules, enabling the formation of diverse and complex structures at the nanoscale.

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.

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.

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 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., \(x + 0 = x\)).

An irrational number is a type of real number that cannot be expressed as a simple fraction or ratio of two integers. This means that if a number is irrational, it cannot be written in the form \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \). Irrational numbers have non-repeating, non-terminating decimal expansions. This means their decimal representations go on forever without repeating a pattern.

Indefinite and fictitious numbers refer to concepts in different mathematical contexts, though they aren't standard terms in a traditional mathematical sense. However, here’s a breakdown of how these terms can be understood: ### Indefinite Numbers Indefinite numbers may refer to numbers that are not fixed or clearly defined.

Pentation is a mathematical operation that is part of the family of hyperoperations, which extend beyond exponentiation. Hyperoperations are defined in a sequence where each operation is one rank higher than the previous one, starting from addition, multiplication, exponentiation, and moving on to tetration and beyond. The sequence is as follows: 1. Addition (a + b) 2. Multiplication (a × b) 3. Exponentiation (a^b) 4.

Sundanese numerals are the number system used by the Sundanese people of West Java, Indonesia. The Sundanese language has its own distinct set of numerals which are used in everyday counting, commerce, and cultural expressions. Here are the Sundanese numerals from one to ten: 1. Satu (1) 2. Dua (2) 3. Tilu (3) 4. Opat (4) 5. Lima (5) 6.

Vote counting is the process of tallying the votes cast in an election to determine the outcome. This process can occur for various types of elections, including political, organizational, or referenda. Here are some key aspects of vote counting: 1. **Methods of Voting**: Votes can be cast in various ways, including in-person on election day, early voting, and absentee or mail-in voting. Each method may involve specific counting protocols.

A list of retired numbers typically refers to the jersey numbers that have been taken out of circulation by a sports team or organization to honor a player, coach, or significant figure associated with that team. When a number is retired, no other player on the team can wear that number in the future. Retired numbers are a common tradition in many sports leagues, including the NFL, NBA, MLB, NHL, and college sports.

A positional numeral system is a method of representing numbers in which the value of a digit depends on its position within a number. In such systems, each position corresponds to a power of a base, and the digits in the number are multiplied by these powers to determine the overall value. ### Key Features of Positional Numeral Systems: 1. **Base**: The base (or radix) of a positional system indicates how many distinct digits (including zero) are available.

The "Cancioneiro de Belém" is a significant collection of Portuguese music that dates back to the early 16th century. It is one of the most important music manuscripts of the Renaissance period in Portugal. The manuscript is named after the Museu de Marinha in Belém, Lisbon, where it is housed. The collection contains a variety of music, including both sacred and secular works, featuring compositions for voices and instruments.

The "Cancionero de Upsala," also known as the "Upsala Songbook," is a significant collection of Spanish poetry from the late 15th century. It contains a variety of songs and secular poetry, primarily written in the Spanish language. The manuscript gained its name from being housed in Uppsala University in Sweden, where it was rediscovered in the 19th century.

The Eton Choirbook is a significant collection of English choral music from the late 15th and early 16th centuries. Compiled around 1500, it contains more than 90 musical works, primarily by English composers of the time, with notable names such as William H. Power, Robert Fayrfax, and Richard Davy represented in the collection.

The Lambeth Choirbook is a significant collection of English choral music from the late 16th century, specifically compiled around 1598. It is named after Lambeth Palace, the London residence of the Archbishop of Canterbury. The choirbook contains a variety of sacred music, primarily composed for the Anglican Church, including settings of the Mass, motets, anthems, and other liturgical works.

Richard Schroeppel is an American computer scientist and mathematician known for his work in the fields of computer science, cryptography, and mathematical logic. He is particularly recognized for his contributions to the development of algorithms and his research in areas such as computational complexity and combinatorial designs. He has also been involved in the exploration of topics related to computer security and cryptographic systems.