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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 ).
All and only those numbers congruent to 0 or 1 (mod 4), except 4, are amenable. (Tamvakis & Lossers 1998) The first few amenable numbers are: 1, 5, 8, 9, 12, 13 ... OEIS: A100832. A solution for integers of the form n = 4k + 1 could be given by a set of 2k (+1)s and 2k (−1)s and n itself. (This generalizes the example of 5 given above.)
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.
In mathematics, an amicable triple is a set of three different numbers so related that the restricted sum of the divisors of each is equal to the sum of other two numbers. [ 1 ] [ 2 ] 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 ...
The aliquot sequence starting with a positive integer k can be defined formally in terms of the sum-of-divisors function σ 1 or the aliquot sum function s in the following way: [1] = = = > = = = If the s n-1 = 0 condition is added, then the terms after 0 are all 0, and all aliquot sequences would be infinite, and we can conjecture that all aliquot sequences are convergent, the limit of these ...
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 ...
284 is in the first pair of amicable numbers with 220. That means that the sum of the proper divisors are the same between the two numbers. [2] 284 can be written as a sum of exactly 4 nonzero perfect squares. [3] 284 is a nontotient number which are numbers where phi(x) equaling that number has no solution. [4] 284 is a number that is the nth ...
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 ...