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In mathematics, the amicable numbers are two different natural numbers related in such a way that the sum of the proper divisors of each is equal to the other number. That is, s ( a )= b and s ( b )= a , where s ( n )=σ( n )- n is equal to the sum of positive divisors of n except n itself (see also divisor function ).
(This generalizes the example of 5 given above.) Although not obvious from the definition, the set of amenable numbers is closed under multiplication (the product of two amenable numbers is an amenable number).
The period of the sequence, or order of the set of sociable numbers, is the number of numbers in this cycle. If the period of the sequence is 1, the number is a sociable number of order 1, or a perfect number—for example, the proper divisors of 6 are 1, 2, and 3, whose sum is again 6. A pair of amicable numbers is a set of sociable numbers of ...
A number that is not part of any friendly pair is called solitary. The abundancy index of n is the rational number σ(n) / n, in which σ denotes the sum of divisors function. A number n is a friendly number if there exists m ≠ n such that σ(m) / m = σ(n) / n. Abundancy is not the same as abundance, which is defined as σ(n) − 2n.
This category consists of titles that consist of numbers.. Use {{}} to add "ambiguous number" disambiguation pages to this category.. When other kinds of things are listed in the disambiguation page, then instead of {{numberdis}}, add {{disambiguation|number|...}} to the disambiguation page (with any other applicable disambiguation categories in additional parameters after |number; see ...
In another equivalent characterization, an amicable triple is a set of three different numbers so related that the sum of the divisors of each is equal to the sum of the three numbers. So a triple ( a , b , c ) of natural numbers is called amicable if s ( a ) = b + c , s ( b ) = a + c and s ( c ) = a + b , or equivalently if σ( a ) = σ( b ...
An example of a probable prime of this form is 200 262144 + 119 262144 (found by Kellen Shenton). [16] By analogy with the ordinary Fermat numbers, it is common to write generalized Fermat numbers of the form + as F n (a). In this notation, for instance, the number 100,000,001 would be written as F 3 (10).
Quasi-sociable numbers or reduced sociable numbers are numbers whose aliquot sums minus one form a cyclic sequence that begins and ends with the same number. They are generalizations of the concepts of betrothed numbers and quasiperfect numbers. The first quasi-sociable sequences, or quasi-sociable chains, were discovered by Mitchell Dickerman ...
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