Arithmetic dynamics is a field of mathematics that combines elements of number theory and dynamical systems. It primarily studies the behavior of sequences of numbers defined by iterative processes, especially those arising from polynomial or rational functions.
An aliquot sequence is a mathematical sequence that begins with a positive integer and continues by repeatedly taking the sum of its proper divisors (the divisors excluding the number itself). Proper divisors are the numbers that divide the original number evenly, apart from the number itself. The sequence can be described as follows: 1. Start with a positive integer \( n \). 2. Find the proper divisors of \( n \) and sum them to get a new number \( a_1 \).
The aliquot sum of a positive integer is the sum of its proper divisors, which are the divisors of the number excluding the number itself. For example, for the number 12, the proper divisors are 1, 2, 3, 4, and 6. Therefore, the aliquot sum of 12 is: 1 + 2 + 3 + 4 + 6 = 16.
A Dudeney number is a special kind of integer that is both a perfect cube and the sum of its digits equals the cube root of the number itself. In mathematical terms, a Dudeney number \( n \) satisfies the condition: \[ n = a^3 \quad \text{and} \quad \text{sum of the digits of } n = a \] where \( a \) is a positive integer.
A **factorion** is a special type of number in mathematics that is equal to the sum of the factorials of its digits. In other words, a number \( n \) is a factorion if: \[ n = d_1! + d_2! + d_3! + \ldots + d_k! \] where \( d_1, d_2, \ldots, d_k \) are the digits of \( n \), and \( !
A **Happy Number** is defined as a number that eventually reaches 1 when replaced repeatedly by the sum of the squares of its digits. If it does not reach 1, it will enter a cycle that does not include 1, and it is then considered an unhappy number. The process for determining if a number is happy can be described as follows: 1. Take the number and replace it with the sum of the squares of its digits. 2. Repeat this process.
A Kaprekar number is a special kind of number in recreational number theory. A non-negative integer \( n \) is called a Kaprekar number if the following condition holds: 1. Square the number \( n \) (calculating \( n^2 \)). 2. Split the resulting square into two parts: the right part containing \( d \) digits (where \( d \) is the number of digits in \( n \)), and the left part containing the remaining digits.
A Keith number is a type of integer that relates to sequences derived from the digits of a number. Given a positive integer \( n \), it is represented in its decimal form. The digits of \( n \) are used to create a sequence where the first terms are derived from the digits of \( n \) and each subsequent term is the sum of the last \( d \) terms, where \( d \) is the number of digits in \( n \).
A Lychrel number is a natural number that is not known to form a palindrome through the iterative process of reversing its digits and adding the result to the original number. A number is considered a palindrome if it reads the same forwards and backwards (for example, 121 or 12321). The Lychrel process typically involves the following steps: 1. Take a natural number n. 2. Reverse its digits to get a new number. 3. Add the reversed number to the original number.
The Meertens number is a concept from the field of programming languages and functional programming, specifically associated with the study of programming language semantics. Named after the Dutch computer scientist Lennart van Hirtum Meertens, it is used to characterize programming languages based on their expressiveness and elegance.
A **multiply perfect number** is a specific type of natural number that can be described in terms of its divisors. Specifically, a natural number \( n \) is called a \( k \)-multiply perfect number if the sum of its divisors (including \( n \) itself), denoted as \( \sigma(n) \), is equal to \( k \) times the number itself.
A Narcissistic number, also known as a pluperfect digital invariant (PDI), is a number that is equal to the sum of its own digits each raised to the power of the number of digits. In simpler terms, for a number \( n \), it can be expressed as: \[ n = d_1^p + d_2^p + d_3^p + ... + d_k^p \] where \( d_1, d_2, ...
A Perfect Digit-to-Digit Invariant (PDDI) is a property of certain types of number transformations that maintain certain characteristics while altering their form. Specifically, it typically refers to a function or operation that transforms a number or a sequence of digits in such a way that each digit in the input number corresponds directly to a digit in the output through a specific relationship.
A **Perfect Digital Invariant (PDI)** is a concept often discussed in the field of cryptography and computer security, particularly in relation to password storage and authentication processes. In a general sense, a PDI refers to a transformation of sensitive information (like a password) such that it remains secure even when it undergoes a number of operations. This concept exemplifies the principle that, despite the operations applied, the invariant retains certain properties that make the original information difficult to derive or reconstruct.
A self number (also known as a Belgian number or a non-generable number) is an integer that cannot be expressed as the sum of a positive integer and the sum of its digits. In other words, a number \( n \) is a self number if there is no positive integer \( m \) such that: \[ n = m + S(m) \] where \( S(m) \) represents the sum of the digits of \( m \).
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