Dirichlet series inversion is a method in analytic number theory that relates Dirichlet series and arithmetic functions. A Dirichlet series is a series of the form \[ F(s) = \sum_{n=1}^\infty \frac{a(n)}{n^s} \] where \( a(n) \) represents a sequence of complex numbers and \( s \) is a complex variable.
A **divisibility sequence** is a sequence of integers \( (a_n) \) where each term divides the subsequent terms in the sequence according to specific criteria. More formally, a sequence \( (a_n) \) is called a divisibility sequence if for each pair of indices \( m < n \), the term \( a_m \) divides \( a_n \) (denoted as \( a_m \mid a_n \)).
The double factorial, denoted by \( n!! \), is a mathematical operation that is defined for non-negative integers. It is the product of all the integers from \( n \) down to 1 that have the same parity (odd or even) as \( n \). Specifically, it is defined as follows: 1. For an even integer \( n = 2k \): \[ n!! = 2k!!
A doubly triangular number is a figurate number that represents a triangular pyramid. In mathematical terms, a doubly triangular number can be derived by summing triangular numbers. The \(n\)-th triangular number \(T_n\) is given by the formula: \[ T_n = \frac{n(n + 1)}{2} \] Doubly triangular numbers can also be expressed in a closed formula.
An elliptic divisibility sequence (EDS) is a sequence of integers that arises from the theory of elliptic curves and has interesting divisibility properties. These sequences are generated based on the coordinates of points on an elliptic curve, typically given in Weierstrass form. The properties of EDSs are linked to the arithmetic of elliptic curves, particularly their group structure.
The Erdős–Nicolas number is a concept from combinatorial number theory that is associated with a particular type of partitioning of the natural numbers. Specifically, it's named after the mathematicians Paul Erdős and Michel Nicolas, who studied certain properties of numbers and sequences.
An Erdős–Woods number is a specific type of integer related to the study of sets of consecutive integers and their relationships to prime numbers. More formally, an integer \( n \) has an Erdős–Woods number if there exists a positive integer \( k \) such that the set of integers: \[ \{ n, n+1, n+2, \ldots, n+k \} \] contains \( k \) or more prime numbers.
An ergodic sequence typically refers to a sequence of random variables or a time series in the context of ergodic theory, which is a branch of mathematics and statistical mechanics. In simple terms, a sequence (or process) is said to be ergodic if, over a long period of time (or a large sample size), its time averages converge to the same value as its ensemble averages.
An Euclid number is a specific type of number that is defined in the context of number theory, particularly concerning prime numbers. The \( n \)-th Euclid number is defined as the product of the first \( n \) prime numbers plus one.
The Euclid–Mullin sequence is a specific sequence of prime numbers that is generated through a recursive process. It starts with the initial prime number 2, and subsequent terms are formed based on the smallest prime that divides the product of all previously generated terms plus one. Here’s how it is generated: 1. Start with \( a_1 = 2 \).
Euler numbers are a sequence of integers that arise in various areas of mathematics, particularly in combinatorics and analysis. There are two main contexts in which the term "Euler numbers" is used: 1. **Euler's Number:** Often referred to as \( e \), this is a fundamental constant in mathematics approximately equal to 2.71828.
Eulerian numbers, denoted as \( E(n, k) \), are a set of integers that count the number of permutations of \( n \) elements in which exactly \( k \) elements appear in ascents. An ascent in a permutation is a position where the next element is larger than the current one.
An **Evil number** is a non-negative integer that has an even number of 1s in its binary representation. For example, the decimal number 3, which is represented in binary as `11`, has two 1s, thus making it an Evil number. In contrast, the number 5, which has a binary representation of `101`, has three 1s and is therefore not an Evil number.
The term "exponential factorial" is not widely used in standard mathematical literature. However, it typically refers to a function that grows extremely quickly, related to the factorial function. Depending on the context, it could imply different things. Here are a couple of interpretations: 1. **Factorial of a Factorial**: One way to interpret "exponential factorial" is to consider the factorial of a factorial, denoted as \( n!
A Fermat number is a specific type of integer that can be expressed in the form: \[ F_n = 2^{2^n} + 1 \] where \( n \) is a non-negative integer. Fermat numbers were named after Pierre de Fermat, a French mathematician, who studied these numbers in the 17th century.
The term "Fermi-Dirac prime" refers to a specific type of prime number that arises from the Fermi-Dirac distribution, which is a statistical distribution that describes the occupancy of energy levels by fermions (particles that follow the Pauli exclusion principle, such as electrons). In more detail, the Fermi-Dirac distribution is used in quantum statistics to describe how particles occupy quantum states at thermal equilibrium, especially at absolute zero temperature.