ALIQUOT SEQUENCE Meaning and
Definition
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An aliquot sequence is a series of positive integers generated using the following algorithm: for each term in the sequence, the term is derived by summing the proper divisors of the previous term. Proper divisors of a positive integer n are any positive integers less than n that divide it evenly, leaving no remainder.
The initial term of an aliquot sequence is typically a positive integer, and subsequent terms are calculated by summing its proper divisors. The resulting sum becomes the next term, and the process is repeated to generate an infinite or finite sequence of numbers.
Aliquot sequences have varying properties and behaviors. For some initial terms, the sequence may quickly converge to a repeating cycle of values. In certain cases, the sequence may eventually reach an integer that marks the end of the sequence. This integer is called a perfect number. Other aliquot sequences might exhibit chaotically increasing or decreasing terms, without reaching a repeating cycle or terminating.
Aliquot sequences have been studied extensively in number theory and recreational mathematics. They offer intriguing patterns and insights into the behavior of integers, divisors, and sums. The study of aliquot sequences often involves exploring the distribution, convergence, and recurrence properties of the terms in the sequence, as well as investigating patterns arising from different initial terms.
Etymology of ALIQUOT SEQUENCE
The word "aliquot" traces back to the Latin word "aliquot", which means "some" or "several". It is derived from the Latin word "aliquotiens", which indicates "several times".
In mathematics, an "aliquot" refers to a part or a fraction of a whole number that does not leave a remainder when divided into it. For example, the aliquot parts of 6 are 1, 2, and 3, as they can evenly divide 6 without leaving a remainder.
In the context of an "aliquot sequence", the term "aliquot" signifies the individual numbers in the sequence that are aliquot parts or divisors of a given number. An aliquot sequence is a sequence of numbers obtained by repeatedly calculating the sum of the aliquot parts of each number in the sequence.
