Misplaced Pages

This article relies largely or entirely on a single source. Relevant discussion may be found on the talk page. Please help improve this article by introducing citations to additional sources. Find sources: "Amicable triple" – news · newspapers · books · scholar · JSTOR (January 2019)

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.

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) = σ(c) = a + b + c. Here σ(n) is the sum of all positive divisors, and s(n) = σ(n) − n is the aliquot sum.

References

1. Dickson, L. E. (1913-03-01). "Amicable Number Triples". The American Mathematical Monthly. 20 (3): 84–92. doi:10.1080/00029890.1913.11997926. ISSN 0002-9890.
2. Dickson, L. E. (1913). "Amicable Number Triples". The American Mathematical Monthly. 20 (3): 84–92. doi:10.2307/2973442. ISSN 0002-9890. JSTOR 2973442.
3. Mason, Thomas E. (1921). "On Amicable Numbers and Their Generalizations". The American Mathematical Monthly. 28 (5): 195–200. doi:10.2307/2973750. ISSN 0002-9890. JSTOR 2973750.

v t e Divisibility-based sets of integers
Overview	Integer factorization Divisor Unitary divisor Divisor function Prime factor Fundamental theorem of arithmetic
Factorization forms	Prime Composite Semiprime Pronic Sphenic Square-free Powerful Perfect power Achilles Smooth Regular Rough Unusual
Constrained divisor sums	Perfect Almost perfect Quasiperfect Multiply perfect Hemiperfect Hyperperfect Superperfect Unitary perfect Semiperfect Practical Descartes Erdős–Nicolas
With many divisors	Abundant Primitive abundant Highly abundant Superabundant Colossally abundant Highly composite Superior highly composite Weird
Aliquot sequence-related	Untouchable Amicable (Triple) Sociable Betrothed
Base-dependent	Equidigital Extravagant Frugal Harshad Polydivisible Smith
Other sets	Arithmetic Deficient Friendly Solitary Sublime Harmonic divisor Refactorable Superperfect

• Divisor function
• Integer sequences
• Number theory