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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 ).
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.) Although not obvious from the definition, the set of amenable numbers is closed under multiplication (the ...
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 of ...
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 ...
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 ...
Every pair a, b of friendly numbers gives rise to a positive proportion of all natural numbers being friendly (but in different clubs), by considering pairs na, nb for multipliers n with gcd(n, ab) = 1. For example, the "primitive" friendly pair 6 and 28 gives rise to friendly pairs 6n and 28n for all n that are congruent to 1, 5, 11, 13, 17 ...
[1] 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 ...
Such a number is algebraic and can be expressed as the sum of a rational number and the square root of a rational number. Constructible number: A number representing a length that can be constructed using a compass and straightedge. Constructible numbers form a subfield of the field of algebraic numbers, and include the quadratic surds.