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.

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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.

Scope of Application¶

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.

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.

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.

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. 5.

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.

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