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Amicable triple

Three distinct natural numbers whose aliquot sum for each equals the sum of the other two, equivalently sharing a divisor sum equal to the triple’s total.

Version
v1 · 2026-09-28 · History
Domain-specific #
7939
Domain group
Formal Sciences
Origin domain
Mathematics
Subdomains
Number Theory, Recreational Mathematics → Mathematics

Core Idea

An amicable triple is a set of three distinct natural numbers (a,b,c) whose proper-divisor sums satisfy (s(a)=b+c), (s(b)=a+c), and (s©=a+b). Equivalently, their full divisor sums all equal the total (a+b+c). Using the full divisor-sum function (\sigma(n)=s(n)+n), those equations are equivalent to (\sigma(a)=\sigma(b)=\sigma©=a+b+c). Using the full divisor-sum function (\sigma(n)=s(n)+n), those equations are equivalent to (\sigma(a)=\sigma(b)=\sigma©=a+b+c).

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Three Number Friends

Every number has smaller numbers that fit into it evenly, like 1, 2 and 3 fit into 6. An amicable triple is three different numbers that are friends in a special way: for each one, its fitting-in numbers add up to exactly the other two numbers put together.

Divisor-Sum Friend Trio

A proper divisor of a number is a smaller number that divides it evenly; the proper divisors of 12 are 1, 2, 3, 4 and 6. An amicable triple is a group of three different numbers where, for each number, the sum of its proper divisors equals the sum of the other two numbers. It's like amicable pairs, where two numbers each have divisor sums equal to the other, but with three friends instead of two. One reported example is 1980, 2016 and 2556. You can check it by finding all their divisors and adding them up.

Three-Way Divisor-Sum Relation

An amicable triple is a set of three distinct natural numbers a, b and c connected by their proper-divisor sums. Write s(n) for the sum of the positive divisors of n that are less than n. The triple must satisfy s(a) = b + c, s(b) = a + c, and s(c) = a + b. Equivalently, using σ(n), the sum of all divisors including n itself, the conditions become σ(a) = σ(b) = σ(c) = a + b + c; the equality to the total of the triple is essential, not just equality among the three. This extends the idea of amicable pairs, in which s(a) = b and s(b) = a, but it is not the same as a chain where each number's divisor sum leads to the next. The smallest reported example is 1980, 2016 and 2556.

 

An amicable triple is a set of three distinct natural numbers (a, b, c) such that s(a) = b + c, s(b) = a + c and s(c) = a + b, where s(n) is the sum of proper divisors of n. Since σ(n) = s(n) + n, these conditions are equivalent to σ(a) = σ(b) = σ(c) = a + b + c, which gives a compact verification, though the equality to the triple's total, not just to each other, is essential. The notion generalizes the reciprocal divisor-sum relation of amicable pairs, but it is not a sequential aliquot cycle in which each number's divisor sum yields the next. The smallest triple reported in the source is (1980, 2016, 2556); factoring 1980 = 2²·3²·5·11, 2016 = 2⁵·3²·7 and 2556 = 2²·3²·71 gives σ = 6552 for each, which equals their sum.

Scope of Application

The definition applies in recreational and computational number theory to finite configurations governed by divisor-sum identities. The definition applies to exact number-theoretic verification, construction, and search for three-member divisor-sum configurations.

  • Verification. Factorization yields σ and s values for a candidate triple.
  • Search. Algorithms constrain common divisor sums and enumerate candidates.
  • Generalized amicability. Triples compare with amicable pairs and larger tuples.
  • Multiplicative number theory. The multiplicativity of σ assists construction and bounding.
  • Mathematical databases. Canonical ordering avoids recording permutations as new triples.

Clarity

Declare natural-number and distinctness conventions, define σ and s, and report all three equations or the full equivalent equality. Canonically sort the members. Do not call a triple amicable from equal divisor sums unless the common value equals its total. The closest near miss sets the boundary: An amicable pair is the closest near miss: each of two numbers has aliquot sum equal to the other, while a triple requires each aliquot sum to equal the sum of two companions.

Manages Complexity

The equivalent σ formulation compresses three complement equations into one common-total test. Factorization decomposes large divisor calculations into prime-power contributions, enabling reliable verification and more efficient search. The central compact equivalence–verification completeness tradeoff is this: Equal σ-values look sufficient but omit the common-total requirement. A second large search space–arithmetic structure tension matters because Brute-force triples are expensive while factorization and σ grouping give leverage.

Abstract Reasoning

Use three linked moves: check that the three entries are distinct natural numbers and sort them canonically; factor each number and compute σ from its prime powers; subtract each number to obtain its aliquot sum s. As a collapse test, the case exits when distinctness fails or any one of the three defining equalities is false. A fourth check is to verify all three complement equations or the common-total equivalent.

Knowledge Transfer

The complement-sum pattern transfers to amicable tuples, but the term triple fixes arity three and the divisor-sum function. Generic mutual support among three objects is only metaphor. No canonical parent prime is currently asserted; broader structural comparisons remain related-prime analogies until separately adjudicated in the DAG. Two formula systems express the same defining relation. Permuting members preserves every condition.

Neighborhood in Abstraction Space

Amicable triple sits in a moderately populated region (55th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Abstract Algebra & Category Theory (24 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08