SOCIABLE NUMBERS Meaning and
Definition
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Sociable numbers are a concept in mathematics that can be defined as a set of positive integers that exhibit a particular relationship with one another through a repetitive pattern of aliquot sums. An aliquot sum refers to the sum of the proper divisors of a given number, excluding the number itself.
More specifically, a set of numbers is considered sociable if each number in the set can be expressed as the aliquot sum of the other numbers within the set. In other words, the aliquot sum of any number within the set will equal another number in the set, thus forming a closed loop of aliquot sums. This cyclic behavior continues indefinitely, with each number eventually returning to its original aliquot sum.
For example, let's consider the sociable set {220, 284}. The aliquot sum of 220 (proper divisors 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110) is 284, and the aliquot sum of 284 (proper divisors 1, 2, 4, 71, and 142) is 220. Therefore, they form a sociable pair.
These sociable numbers are an intriguing field of study in mathematics, as they exhibit fascinating patterns and have connections to various branches of mathematics, including number theory and algebraic graph theory. They have been a subject of investigation and curiosity for mathematicians seeking to understand the properties and behavior of these unique sets of numbers.
Etymology of SOCIABLE NUMBERS
The word "sociable" originates from the Latin word "sociabilis", which means "friendly" or "companionable". It was derived from the Latin word "socius", meaning "companion" or "ally". The term "sociable numbers" itself refers to a mathematical concept related to the classification of numbers, which was likely named "sociable" due to the interaction or relationship that these numbers exhibit.
