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Main article: Amicable numbers § Amicable tuples
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 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.[3]
^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.
^Dickson, L. E. (1913). "Amicable Number Triples". The American Mathematical Monthly. 20 (3): 84–92. doi:10.2307/2973442. ISSN 0002-9890. JSTOR 2973442.
^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.
In mathematics, an amicabletriple is a set of three different numbers so related that the restricted sum of the divisors of each is equal to the sum...
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expressed as the sum of its own proper divisors. Similarly, none of the amicable numbers or sociable numbers are untouchable. Also, none of the Mersenne...
sequences eventually end with a prime number, a perfect number, or a set of amicable or sociable numbers? (Catalan's aliquot sequence conjecture) (more unsolved...
the powers of 2. Betrothed numbers relate to quasiperfect numbers like amicable numbers relate to perfect numbers. Hagis, Peter; Cohen, Graeme L. (1982)...
of numbers which are the sum of each other's proper divisors are called amicable, and larger cycles of numbers are called sociable. A positive integer such...
sequence. They are generalizations of the concepts of perfect numbers and amicable numbers. The first two sociable sequences, or sociable chains, were discovered...
binary quadratic forms of discriminant Δ denoted by GΔ. GΔ is the set of triples of integers (a, b, c) in which those integers are relative prime. Given...
Betrothed numbers or quasi-amicable numbers are two positive integers such that the sum of the proper divisors of either number is one more than the value...
MR 2394891, S2CID 3207238, Zbl 1163.11059 Gioia, A. A.; Vaidya, A. M. (1967), "Amicable numbers with opposite parity", The American Mathematical Monthly, 74 (8):...
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