Hooley's delta function, often denoted as \( \Delta(s) \), is a mathematical tool used in number theory, particularly in the context of the Generalized Riemann Hypothesis and the distribution of prime numbers. It was introduced by C. Hooley in his work related to the study of integers represented by quadratic forms and sieve methods. The function is defined in terms of the values of L-functions, specifically for certain Dirichlet series associated with characters.
A home prime is a concept in number theory that relates to the representation of numbers as sums of prime numbers. Specifically, a home prime is produced by repeatedly factoring a composite number into its prime factors, then concatenating those prime factors (written in order), and repeating the process until a prime number is obtained. Here’s how it works in detail: 1. Start with a composite number. 2. Factor it into its prime factors.
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.
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.
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 \( n \).
A highly powerful number is defined as a positive integer \( n \) such that for every prime \( p \) that divides \( n \), \( p^2 \) also divides \( n \). In other words, in the prime factorization of a highly powerful number, each prime factor appears with an exponent of at least 2.
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 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 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 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 **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 \( n \).
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 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.
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.
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.
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.
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 \( n \).
The Golomb sequence is a non-decreasing integer sequence where each positive integer \( n \) appears exactly \( G(n) \) times in the sequence.
A Giuga number is a special type of natural number defined by a property related to prime numbers and their factors.
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!