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Base-dependent integer sequences are sequences of integers that vary based on the numeral system (base) used to represent numbers. In other words, the way we express numbers in different bases can lead to different sequences of integers when applying specific rules or transformations. ### Key Concepts: 1. **Base Representation**: Each integer can be represented in different numeral systems, such as binary (base 2), decimal (base 10), hexadecimal (base 16), etc.

A binary sequence is a sequence of numbers where each number is either a 0 or a 1. These sequences are fundamental in various fields, particularly in computer science and digital electronics, as they represent the most basic form of data storage and processing. ### Characteristics of Binary Sequences: 1. **Composition**: Each element of the sequence can take on one of two possible values: 0 or 1.

Fibonacci numbers are a sequence of numbers in which each number (after the first two) is the sum of the two preceding ones. The sequence starts with 0 and 1, and proceeds as follows: - 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...

A perfect number is a positive integer that is equal to the sum of its proper divisors, excluding itself. In simpler terms, a perfect number is a number that is the sum of its divisors (excluding the number itself). For example: - The first perfect number is 6. Its divisors are 1, 2, and 3, and their sum is \(1 + 2 + 3 = 6\). - The second perfect number is 28.

Pseudoprimes are composite numbers that satisfy certain properties of prime numbers in specific mathematical contexts. More formally, a pseudoprime relates to the concept of prime numbers in that they can pass certain primality tests, which are typically designed to identify prime numbers. One common type of pseudoprime is the "Fermat pseudoprime.

An **abundant number** is a positive integer for which the sum of its proper divisors (the positive divisors excluding the number itself) is greater than the number itself.

An Achilles number is a positive integer that is a powerful number but not a perfect power. A powerful number is defined as a number \( n \) such that in its prime factorization, every prime number \( p \) appears with an exponent of at least 2. In contrast, a perfect power is a number of the form \( m^k \) where \( m \) and \( k \) are positive integers and \( k \geq 2 \).

Alcuin's sequence is a sequence of numbers that begins with 1 and follows a specific pattern defined by a recurrence relation associated with the mathematician Alcuin of York. The sequence is often described as follows: - The first term \( a_0 = 1 \).

An "almost perfect number" is a type of natural number that is closely related to perfect numbers. A perfect number is a positive integer that is equal to the sum of its proper divisors (excluding itself). For example, 6 is a perfect number because its divisors (1, 2, and 3) add up to 6.

An "almost prime" is a term often used in number theory to refer to natural numbers that have a specific number of prime factors. The most common interpretation is that an almost prime is a positive integer that has exactly \( k \) prime factors, counting multiplicities. For example: - If \( k = 1 \), then the almost primes are the prime numbers themselves (like 2, 3, 5, 7, etc.

The concept of alternating factorial refers to a specific way of calculating a factorial that alternates the signs of the terms. For a non-negative integer \( n \), the alternating factorial \( !n \) is defined as follows: \[ !

Amicable numbers are a pair of numbers for which the sum of the proper divisors (factors excluding the number itself) of each number equals the other number. In other words, if you have two numbers, \(A\) and \(B\), they are considered amicable if: 1. The sum of the proper divisors of \(A\) (denoted as \(σ(A) - A\)) equals \(B\).

An amicable triple is a generalization of the concept of amicable numbers. While amicable numbers are two different integers where each number is the sum of the proper divisors of the other, an amicable triple consists of three different integers \( (a, b, c) \) such that the sum of the proper divisors of each integer equals the sum of the other two.

An arithmetic number is not a standard term widely recognized in mathematics, but it could refer to different concepts depending on the context. Here are a couple of interpretations: 1. **Arithmetic Sequences**: In the context of sequences, an arithmetic number could refer to the numbers in an arithmetic sequence, which is a sequence of numbers in which the difference between any two consecutive terms is constant. For example, in the sequence 2, 5, 8, 11, ...

An arithmetico-geometric sequence is a sequence in which each term is generated by multiplying an arithmetic sequence by a geometric sequence. In simple terms, it combines the elements of arithmetic sequences (which have a constant difference between consecutive terms) and geometric sequences (which have a constant ratio between consecutive terms).

An **automatic sequence** is a type of numerical sequence that is generated by a specific rule or algorithm, often involving a function or a set of operations that can be repeated indefinitely. The defining characteristic of an automatic sequence is that it can be described by a finite automaton, which means that given any input (usually an integer representing the position in the sequence), the automaton can produce the corresponding term in the sequence without the need for memory of past values.

In set theory and topology, a **Baire space** is a topological space that satisfies a particular property related to the concept of "largeness" in topology. Specifically, a topological space \( X \) is called a Baire space if the intersection of any countable collection of dense open sets in \( X \) is dense in \( X \).

The term "Ban number" can refer to different concepts depending on the context, and it is not a widely recognized standard term. 1. **Legal Context**: In some legal contexts, a ban number could refer to a case or legal action identifier assigned to a specific prohibition or restriction. 2. **Telecommunications**: In some telecommunications circles, "BAN" might refer to a "Billing Account Number," which is used to identify a customer's billing account.

The Beatty sequence is a sequence of numbers that can be derived from the mathematical concept of filling the real line with two sequences whose terms are the floor functions of the multiples of two irrational numbers.

The Behrend sequence refers to a construction in combinatorial number theory that produces sets of integers with certain properties related to the sum of their elements. In particular, the Behrend sequence is often associated with sets of integers that do not contain three-term arithmetic progressions.