Aliquot sum¶

In number theory, the aliquot sum of a positive integer is the sum of all proper divisors of , that is, all divisors of other than itself.

Version: v1 · 2026-09-28 · History
Domain-specific #: 7924
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomain: Number Theory → Mathematics

Core Idea¶

Aliquot sum is treated here as the recurring formal models and representations identity summarized by this source-grounded definition: In number theory, the aliquot sum of a positive integer is the sum of all proper divisors of , that is, all divisors of other than itself.

In number theory, the aliquot sum of a positive integer is the sum of all proper divisors of , that is, all divisors of other than itself. For example, the proper divisors of 12 (that is, the positive divisors of 12 that are not equal to 12) are , and 6, so the aliquot sum of 12 is 16 i.e. Sociable numbers are numbers whose aliquot sequence is a periodic sequence.

A number is prime if and only if its aliquot sum is 1. The aliquot sums of perfect, deficient, and abundant numbers are equal to, less than, and greater than the number itself respectively. The quasiperfect numbers (if such numbers exist) are the numbers whose aliquot sums equal .

For Aliquot sum, the abstraction is narrower than the article's general subject matter: a positive case must preserve In number theory, the aliquot sum of a positive integer is the sum of all proper divisors of , that is, all divisors of other than itself. Retaining only the name, a familiar example, or a downstream effect is insufficient. The specialist roles and tests remain anchored in formal models and representations, which is why this identity is domain-specific rather than prime.

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The Smaller-Helpers Sum

Take a number, like 12, and find all the smaller numbers that fit into it evenly: 1, 2, 3, 4, and 6. Add them up and you get 16. That total is called the aliquot sum of 12.

Sum of Smaller Divisors

The aliquot sum of a whole number is what you get by adding up all the numbers that divide it evenly, except the number itself. For 12, those are 1, 2, 3, 4, and 6, which add to 16. A number is prime exactly when its aliquot sum is 1, because only 1 divides it besides itself. If the aliquot sum equals the number, like 6 = 1 + 2 + 3, the number is called perfect. If the sum is smaller, the number is deficient, and if it's bigger, like 12, it's abundant.

Sum of Proper Divisors

In number theory, the aliquot sum s(n) of a positive integer n is the sum of its proper divisors, meaning all its positive divisors except n itself. For example, 12 has proper divisors 1, 2, 3, 4, and 6, so s(12) = 16. The aliquot sum sorts numbers: n is perfect if s(n) = n, deficient if s(n) < n, and abundant if s(n) > n. A number is prime exactly when its aliquot sum is 1. Quasiperfect numbers would have s(n) = n + 1, though no one knows if any exist. Applying s over and over gives an aliquot sequence, and numbers whose sequences cycle back on themselves are called sociable numbers.

The aliquot sum s(n) of a positive integer n is the sum of its proper divisors, that is, all positive divisors except n itself; equivalently, s(n) = σ(n) − n, where σ is the sum of all divisors. For example, s(12) = 1 + 2 + 3 + 4 + 6 = 16. The number n is prime if and only if s(n) = 1. Perfect, deficient, and abundant numbers are those with s(n) equal to, less than, and greater than n, respectively. Quasiperfect numbers, if any exist, would satisfy s(n) = n + 1. Iterating s produces the aliquot sequence of n, and sociable numbers are those whose aliquot sequence is periodic. The identity requires summing exactly the proper divisors; including n or restricting to prime divisors gives a different function.

Structural Signature¶

Sig role-phrases:

Defining carrier — It can be used to characterize the prime numbers, perfect numbers, sociable numbers, deficient numbers, abundant numbers, and untouchable numbers, and to define the aliquot sequence of a number.
Constitutive relation — For example, the proper divisors of 12 (that is, the positive divisors of 12 that are not equal to 12) are , and 6, so the aliquot sum of 12 is 16 i.e.
Operating condition — The aliquot sum function can be used to characterize several notable classes of numbers.
Recognition evidence — A number is prime if and only if its aliquot sum is 1.
Admissible variation — The aliquot sums of perfect, deficient, and abundant numbers are equal to, less than, and greater than the number itself respectively.
Characteristic consequence — The quasiperfect numbers (if such numbers exist) are the numbers whose aliquot sums equal .
Failure boundary — The almost perfect numbers (which include the powers of 2, being the only known such numbers so far) are the numbers whose aliquot sums equal .

What It Is Not¶

Not the whole field of formal models and representations. The node requires the specific identity stated by In number theory, the aliquot sum of a positive integer is the sum of all proper divisors of , that is, all divisors of other than itself.
Not an over-broad reading. For example, the proper divisors of 12 (that is, the positive divisors of 12 that are not equal to 12) are , and 6, so the aliquot sum of 12 is 16 i.e.
Not an over-broad reading. The untouchable numbers are the numbers that are not the aliquot sum of any other number.
Not an over-broad reading. It can be used to characterize the prime numbers, perfect numbers, sociable numbers, deficient numbers, abundant numbers, and untouchable numbers, and to define the aliquot sequence of a number.
Not automatically Untouchable Number. Retrieval proximity does not establish equivalence; the two identities must be compared by carrier, operation, and failure boundary.

Scope of Application¶

Aliquot sum applies literally inside formal models and representations wherever the source-defined carrier and relation can be established. Its documented habitats include:

Characterization of classes of numbers. The aliquot sum function can be used to characterize several notable classes of numbers.
That is,. It can be used to characterize the prime numbers, perfect numbers, sociable numbers, deficient numbers, abundant numbers, and untouchable numbers, and to define the aliquot sequence of a number.
Characterization of classes of numbers. The mathematicians noted that one of Erdős' "favorite subjects of investigation" was the aliquot sum function.
Iteration. Iterating the aliquot sum function produces the aliquot sequence of a nonnegative integer (in this sequence, we define ).
Examples. For example, the proper divisors of 12 (that is, the positive divisors of 12 that are not equal to 12) are , and 6, so the aliquot sum of 12 is 16 i.e.
Characterization of classes of numbers. A number is prime if and only if its aliquot sum is 1.

Outside formal models and representations, the name should be retained only when these same operational conditions survive; otherwise the comparison belongs to the broader parent Theory or should be marked as analogy.

Clarity¶

A clear use of Aliquot sum names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In number theory, the aliquot sum of a positive integer is the sum of all proper divisors of , that is, all divisors of other than itself. The strongest recognition evidence in the frozen account is: A number is prime if and only if its aliquot sum is 1. A report should distinguish that evidence from a proxy, consequence, or common implementation. It should also state the qualification For example, the proper divisors of 12 (that is, the positive divisors of 12 that are not equal to 12) are , and 6, so the aliquot sum of 12 is 16 i.e. so that a reader can reproduce the classification rather than infer it from topical resemblance.

Manages Complexity¶

Aliquot sum compresses multiple formal models and representations details into a stable diagnostic relation. The source shows both the central mechanism—for example, the proper divisors of 12 (that is, the positive divisors of 12 that are not equal to 12) are , and 6, so the aliquot sum of 12 is 16 i.e.—and the practical consequence—the quasiperfect numbers (if such numbers exist) are the numbers whose aliquot sums equal . This compression makes cases comparable while leaving parameters, conventions, exceptions, and evidential quality explicit. It is lossy by design: local history and implementation details may be omitted only when they do not alter the defining relation.

Abstract Reasoning¶

Type the carrier. Identify the formal models and representations entities to which the claim applies.
State the relation. Use the source-grounded identity: In number theory, the aliquot sum of a positive integer is the sum of all proper divisors of , that is, all divisors of other than itself.
Check operation and conditions. The aliquot sum function can be used to characterize several notable classes of numbers.
Demand recognition evidence. A number is prime if and only if its aliquot sum is 1.
Test variation. Change an implementation or setting while preserving the aliquot sums of perfect, deficient, and abundant numbers are equal to, less than, and greater than the number itself respectively.
Run the collapse test. Remove the defining operation; if the label still seems equally apt, only a topic or correlate was retained.
Reduce cautiously. When the specialist conditions cannot be carried, route the residual comparison to Theory.

Knowledge Transfer¶

Within the home domain. Knowledge about Aliquot sum transfers literally when a new case preserves the same carrier type, relation, and recognition test. The aliquot sum function can be used to characterize several notable classes of numbers. It can be used to characterize the prime numbers, perfect numbers, sociable numbers, deficient numbers, abundant numbers, and untouchable numbers, and to define the aliquot sequence of a number.

Beyond the home domain. No canonical parent is asserted for Aliquot sum. An outside case receives the specialist name only when the same typed roles and rejection conditions can be filled literally; otherwise the comparison remains an analogy pending later graph densification.

Examples¶

Canonical¶

For example, the proper divisors of 12 (that is, the positive divisors of 12 that are not equal to 12) are , and 6, so the aliquot sum of 12 is 16 i.e. This case is canonical because it supplies a concrete carrier and lets the defining relation be checked rather than merely named.

Mapped back: carrier → the entities in the documented case; operation → In number theory, the aliquot sum of a positive integer is the sum of all proper divisors of , that is, all divisors of other than itself; recognition evidence → A number is prime if and only if its aliquot sum is 1

Applied / In Practice¶

The aliquot sum function can be used to characterize several notable classes of numbers. The applied case shows how the identity is used under a second setting or qualification while keeping the same operative relation.

Mapped back: changed setting → Characterization of classes of numbers; invariant → In number theory, the aliquot sum of a positive integer is the sum of all proper divisors of , that is, all divisors of other than itself; boundary → the case exits the class when for example, the proper divisors of 12 (that is, the positive divisors of 12 that are not equal to 12) are , and 6, so the aliquot sum of 12 is 16 i.e

Structural Tensions¶

T1 — Stable identity versus admissible variation. For example, the proper divisors of 12 (that is, the positive divisors of 12 that are not equal to 12) are , and 6, so the aliquot sum of 12 is 16 i.e. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Which changes preserve the defining relation, and which replace it?

T2 — Recognition versus proxy. The untouchable numbers are the numbers that are not the aliquot sum of any other number. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the cited evidence establish the identity or only a correlated sign?

T3 — Definition versus implementation. It can be used to characterize the prime numbers, perfect numbers, sociable numbers, deficient numbers, abundant numbers, and untouchable numbers, and to define the aliquot sequence of a number. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Is the observed implementation constitutive, optional, or merely common?

T4 — Scope versus overextension. The aliquot sum function can be used to characterize several notable classes of numbers. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Can every claimed application fill the same typed roles without metaphor?

T5 — Transfer versus domain accent. It can be used to characterize the prime numbers, perfect numbers, sociable numbers, deficient numbers, abundant numbers, and untouchable numbers, and to define the aliquot sequence of a number. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the receiving case instantiate Aliquot sum literally, co-instantiate Theory, or only resemble it?

T6 — Autonomy versus reduction. For example, the proper divisors of 12 (that is, the positive divisors of 12 that are not equal to 12) are , and 6, so the aliquot sum of 12 is 16 i.e. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: What does Aliquot sum distinguish that the broader parent Theory leaves together?

Structural–Framed Character¶

Aliquot sum is mixed or framed-leaning. Its structural side is the repeatable organization summarized by In number theory, the aliquot sum of a positive integer is the sum of all proper divisors of , that is, all divisors of other than itself. Its framed side is the formal models and representations vocabulary that fixes the carrier, evidence, exceptions, and admissible transformations.

Evaluative weight: the identity can be stated descriptively even when applications carry practical stakes. Human-practice dependence: the source-grounded carrier determines whether the relation exists independently or is constituted by a practice. Institutional origin: disciplinary conventions stabilize the name and test. Vocabulary portability: The aliquot sum function can be used to characterize several notable classes of numbers. Import versus recognition: literal transfer requires the same mechanism; shape alone is analogy.

Its portable skeleton is Theory. Its character: a recurring specialist identity whose thin organization can be abstracted, while its operational meaning remains domain-bound.

Structural Core vs. Domain Accent¶

What is skeletal. In number theory, the aliquot sum of a positive integer is the sum of all proper divisors of , that is, all divisors of other than itself. The stable skeleton is the typed relation expressed in that definition and the entry's recognition and collapse tests. The source identifies these operative conditions: It can be used to characterize the prime numbers, perfect numbers, sociable numbers, deficient numbers, abundant numbers, and untouchable numbers, and to define the aliquot sequence of a number. For example, the proper divisors of 12 (that is, the positive divisors of 12 that are not equal to 12) are , and 6, so the aliquot sum of 12 is 16 i.e. It further constrains recognition and variation through: The aliquot sum function can be used to characterize several notable classes of numbers. A number is prime if and only if its aliquot sum is 1.

What is domain-bound. formal models and representations supplies the operative entities, technical vocabulary, warrants, and exceptions that make Aliquot sum literal. Its documented scope includes the condition that The aliquot sum function can be used to characterize several notable classes of numbers. Another bounded application condition is that It can be used to characterize the prime numbers, perfect numbers, sociable numbers, deficient numbers, abundant numbers, and untouchable numbers, and to define the aliquot sequence of a number. These are not decorative examples; they determine which carrier and evidence can fill the abstraction's roles.

Why no parent is asserted. Removing those specialist details does not currently yield one live catalog node that is a necessary genus for every instance. The entry is therefore approved as unparented rather than attached by topical resemblance. Its collapse evidence remains specific—The aliquot sums of perfect, deficient, and abundant numbers are equal to, less than, and greater than the number itself respectively.—and future graph densification may discover a defensible relation only if it preserves that boundary.

This entry is a kind of Arithmetic function.

Approved unparented node. No current live node supplies a defensible necessary genus or structural prerequisite for Aliquot sum. The reviewed identity is: In number theory, the aliquot sum of a positive integer is the sum of all proper divisors of, that is, all divisors of other than itself. The accelerated suggestion was declined because topical or lexical similarity does not establish hierarchy; the node is admitted without a parent pending later graph densification.
Related reasoning operations. Evidence, representation, comparison, classification, transformation, or evaluation may participate in particular cases, but participation does not make any one of them a necessary parent of every instance.

Relationships to Other Abstractions¶

Current abstraction Aliquot sum Domain-specific

Parents (1) — more general patterns this builds on

Aliquot sum is a kind of Arithmetic function Domain-specific

Aliquot sum is a strict kind of Arithmetic function: its frozen identity entails the parent's defining structure while adding domain-specific restrictions.

Hierarchy path (1) — routes to 1 parentless root

Aliquot sum → Arithmetic function → Function (Mapping)

Neighborhood in Abstraction Space¶

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

Family — Number-Theoretic Properties & Tests (20 abstractions)

Nearest neighbors

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

Not to Be Confused With¶

Theory. The parent omits the specialist differentia. Tell: Can the case establish In number theory, the aliquot sum of a positive integer is the sum of all proper divisors of , that is, all divisors of other than itself?
Untouchable Number. A positive integer lying outside the image of the aliquot-sum function, classified by the nonexistence of any positive integer whose proper divisors sum to it. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Perfect number. Classify a positive integer as perfect when the sum of all its proper positive divisors equals the integer itself, equivalently when its divisor-sum is exactly twice the integer. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Multiply perfect number. A positive integer whose sum of positive divisors is an integer multiple k of the number itself. Tell: Which entry's carrier, operation, and failure condition are satisfied?
A measurement, proxy, or consequence. Those may provide evidence without being the identity. Tell: Would Aliquot sum remain present if the detector or downstream effect changed?
A metaphorical analogue. A similar shape outside formal models and representations lacks the specialist mechanism. Tell: Do the native roles transfer literally, or only the parent Theory?

References¶

Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Aliquot_sum (revision 1363371063).
Preserved source candidate: https://users.renyi.hu/~p_erdos/1973-27.pdf

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.