mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, an amicable triple is a

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

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 number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...

s 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 In number theory, the aliquot sum ''s''(''n'') of a positive integer ''n'' is the sum of all proper divisors of ''n'', that is, all divisors of ''n'' other than ''n'' itself. That is, :s(n)=\sum\nolimits_d. It can be used to characterize the prim ...

References

{{Classes of natural numbers Divisor function Integer sequences Number theory