amicable triple

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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.{{Cite journal|last=Dickson|first=L. E.|date=1913-03-01|title=Amicable Number Triples|url=https://doi.org/10.1080/00029890.1913.11997926|journal=The American Mathematical Monthly|volume=20|issue=3|pages=84–92|doi=10.1080/00029890.1913.11997926|issn=0002-9890|url-access=subscription}}{{Cite journal|last=Dickson|first=L. E.|date=1913|title=Amicable Number Triples|url=https://www.jstor.org/stable/2973442|journal=The American Mathematical Monthly|volume=20|issue=3|pages=84–92|doi=10.2307/2973442|jstor=2973442 |issn=0002-9890|url-access=subscription}}

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.{{Cite journal|last=Mason|first=Thomas E.|date=1921|title=On Amicable Numbers and Their Generalizations|url=https://www.jstor.org/stable/2973750|journal=The American Mathematical Monthly|volume=28|issue=5|pages=195–200|doi=10.2307/2973750|jstor=2973750 |issn=0002-9890|url-access=subscription}}