Amicable triple

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.<ref>{{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}}</ref><ref>{{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}}</ref>

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. The smallest amicable triple is (1980, 2016, 2556). <ref>{{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}}</ref>

== References == <references />{{Divisor classes}} {{Classes of natural numbers}}

Category:Divisor function Category:Integer sequences Category:Number theory