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 fractal sequence is a series of elements that exhibit a recursive or self-similar structure, often characterized by repeating patterns at various scales. In mathematics and specifically in the field of fractal geometry, a fractal is often defined through its property of self-similarity, meaning that parts of the fractal resemble the whole structure.
A "friendly number" typically refers to a number that is part of a pair or set of numbers with a mutual relationship, where two numbers share a specific mathematical characteristic. The term is most commonly associated with the concept of "friendly pairs" or "friendly numbers" in the context of number theory, particularly in relation to amicable numbers.
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 Giuga number is a special type of natural number defined by a property related to prime numbers and their factors.
The Golomb sequence is a non-decreasing integer sequence where each positive integer \( n \) appears exactly \( G(n) \) times in the sequence.
Goodstein's theorem is a result in mathematical logic and number theory that deals with a particular sequence of natural numbers known as Goodstein sequences. The theorem states that every Goodstein sequence eventually terminates at 0, despite the fact that the terms of the sequence can grow extremely large before reaching 0. To understand Goodstein's theorem, we first need to define how a Goodstein sequence is constructed: 1. **Starting Point**: Begin with a natural number \( n \).
Gould's sequence is a sequence of numbers that describes a particular arrangement of integers based on the principle of mathematical games and strategy. Specifically, it is generated using a recursive process related to the game of Nim and other combinatorial games. In his exploration of combinatorial game theory, mathematician Steven Jay Gould defined this sequence as follows: 1. Start with the first term, which is typically 0.
Gregory coefficients, also known as Gregory series coefficients, are used in the context of approximation and numerical analysis, particularly related to interpolation and numerical integration. They are named after the mathematician James Gregory, who made significant contributions to the field of mathematics in the 17th century. In many cases, Gregory coefficients are associated with a specific type of polynomial interpolation called the Gregory-Newton interpolation formula. This formula provides a way to construct an interpolating polynomial based on a set of data points.
Göbel's sequence is an integer sequence defined by a specific recursive relation. It begins with two initial values, often 0 and 1, and subsequent terms are generated based on the values of previous terms in the sequence.
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.
In topology, a **Hausdorff gap** (or just **gap**) is a concept relating to the structure of certain topological spaces, specifically in relation to the properties of sequences or nets in those spaces. A Hausdorff gap is often associated with the concept of a *Hausdorff space*, which is a topological space where any two distinct points can be separated by neighborhoods.
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 Hermite number is a specific kind of number that arises in the context of algebraic number theory and is related to Hermite's work in mathematics. However, the term "Hermite number" is not widely used or standardized in mathematics, and it may not refer to a universally recognized concept.
A highly abundant number is a positive integer that has a particularly high ratio of the sum of its divisors to the number itself. More formally, a highly abundant number \( n \) satisfies the condition that for any integer \( m < n \), the sum of the divisors function \( \sigma(m) \) (which returns the sum of all positive divisors of \( m \)) is less than \( \sigma(n) \) divided by \( n \).
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 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 \( x \).
A Highly Totient Number is a positive integer \( n \) for which the equation \[ \Phi(\Phi(\Phi(... \Phi(n)...))) = 1 \] holds true after applying the Euler's totient function \( \Phi \) repeatedly a positive number of times. The Euler's totient function \( \Phi(n) \) counts the number of positive integers up to \( n \) that are relatively prime to \( n \).
The Hilbert number is generally associated with the concept of the Hilbert space and refers to a specific enumeration of points in such spaces. However, in a more concrete mathematical context, "Hilbert numbers" are often used to refer to certain types of sequences or series associated with the work of the mathematician David Hilbert, particularly in relation to cardinalities, sets, and various hierarchies within mathematical analysis or topology.