Skip to content

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) linked by their proper-divisor sums. Writing (s(n)) for the sum of positive divisors less than (n), the defining conditions are (s(a)=b+c), (s(b)=a+c), and (s©=a+b).

Using the full divisor-sum function (\sigma(n)=s(n)+n), those equations are equivalent to (\sigma(a)=\sigma(b)=\sigma©=a+b+c). The equivalence offers a compact verification, but the equality to the triple’s total is essential.

The relation generalizes the reciprocal divisor-sum idea of amicable pairs without forming a sequential aliquot cycle. The smallest triple reported in the frozen source is (1980,2016,2556); direct factorization and divisor sums verify all three equations.

How would you explain it like I'm…

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.

Structural Signature

Sig role-phrases:

  • Three distinct naturals. Supply the unordered candidate set {a,b,c} with no repeated member. Constitutive carrier. If altered: A pair, larger tuple, repeated value, or non-natural entry is outside the definition.
  • Aliquot-sum function s. Adds proper positive divisors and excludes the number itself. Constitutive arithmetic operation. If altered: Using σ without adjusting by n changes the stated equations.
  • Pairwise complement equations. Require each member’s aliquot sum to equal the other two members combined. Identity-bearing relation. If altered: Satisfying only one or two equations does not form an amicable triple.
  • Common full-divisor total. Reexpresses the three equations as equal σ-values matching a+b+c. Equivalent verification form. If altered: Equal σ-values alone are insufficient if they do not equal the triple total.

What It Is Not

  • Not an amicable pair. Three members and complement sums replace two reciprocal aliquot sums.
  • Not a sociable cycle. The aliquot sum maps to the sum of companions, not to one next member.
  • Not equal σ alone. The common value must equal a+b+c.
  • Not an ordered triple. Permutation does not change the set relation.

Scope of Application

The definition applies in recreational and computational number theory to finite configurations governed by divisor-sum identities.

  • 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.

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.

Abstract Reasoning

  1. Check that the three entries are distinct natural numbers and sort them canonically.
  2. Factor each number and compute σ from its prime powers.
  3. Subtract each number to obtain its aliquot sum s.
  4. Verify all three complement equations or the common-total equivalent.
  5. Use exact arithmetic and preserve factorization evidence for large computational claims.

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.

Examples

Canonical

For (1980, 2016, 2556), exact divisor sums satisfy s(1980)=2016+2556, s(2016)=1980+2556, and s(2556)=1980+2016.

Mapped back: three distinct naturals → 1980, 2016, 2556; aliquot-sum function s → proper-divisor totals; pairwise complement equations → all three equalities; common full-divisor total → σ values equal the triple sum.

Applied / In Practice

A search groups integers by σ(n), chooses three distinct members from one group, and then retains only sets whose member sum equals that common σ-value.

Mapped back: three distinct naturals → candidate three-subsets; aliquot-sum function s → derivable from σ−n; pairwise complement equations → implied after final test; common full-divisor total → search key plus sum filter.

Structural Tensions

T1: compact equivalence vs. verification completeness. Equal σ-values look sufficient but omit the common-total requirement. Diagnostic: Was σ=a+b+c checked?

T2: large search space vs. arithmetic structure. Brute-force triples are expensive while factorization and σ grouping give leverage. Diagnostic: Which constraints prune candidates without losing solutions?

T3: set identity vs. tuple enumeration. Permutations can inflate computational counts. Diagnostic: Is canonical ordering enforced?

Structural–Framed Character

Amicable triple is strongly structural. Evaluative weight: none inherent. Human-practice-bound: natural-number conventions and naming are stipulated. Institutional origin: number theory stabilizes the identity. Vocabulary travels: equivalence and sum relations travel. Import versus recognize: literal use requires exact divisor sums. Its character: a symmetric three-number fixed relation under aliquot complementation.

Structural Core vs. Domain Accent

Skeletal core. Each member’s derived value equals the aggregate of all other members, yielding a symmetric common-total form.

Domain-bound accent. Members are natural numbers and the derived value is the proper-divisor sum.

Why not prime. Symmetric complement relations are portable, but amicable triple is an exact number-theoretic configuration.

  • Equivalence. Two formula systems express the same defining relation.
  • Symmetry. Permuting members preserves every condition.
  • Constraint. All three arithmetic equalities must hold jointly.
  • The approved root remains.

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

Not to Be Confused With

  • Amicable pair. Tell: It contains two reciprocally related numbers.
  • Sociable numbers. Tell: They form an iterative aliquot-sum cycle.
  • Multiamicable numbers. Tell: Those satisfy different multiplier relations for σ.
  • Equal-divisor-sum numbers. Tell: Equal σ-values need not equal the sum of the selected triple.

References

  • Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Amicable_triple (revision 1368971146).
  • Preserved source candidate: https://doi.org/10.1080/00029890.1913.11997926
  • Preserved source candidate: https://www.jstor.org/stable/2973442
  • Preserved source candidate: https://www.jstor.org/stable/2973750

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.