Arithmetic number

A positive integer whose positive divisors have an integer arithmetic mean.

Version

v2 · 2026-09-19 · History

Domain-specific #

3330

Domain group

Formal Sciences

Origin domain

Mathematics

Subdomain

Number Theory → Mathematics

Core Idea

An integer n is arithmetic when its sum-of-divisors value is divisible by its divisor-count value, equivalently when sigma(n) divided by tau(n) is integral. The divisor multiset maps n to two multiplicative-function statistics, and a divisibility test determines whether its average closes in the integers. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

The load-bearing residual is not the broad topic of number theory. It is the domain-specific identity determined by positive divisors are counted once, the arithmetic mean equals sigma(n) over tau(n), and the quotient is an integer.

How would you explain it like I'm…

The Fair-Share Numbers

Take a number, like 6. Find all the numbers that fit into it evenly: 1, 2, 3, and 6. Add them up to get 12, then share that equally among the 4 numbers, and each gets exactly 3 -- no leftovers! A number whose sharing comes out even like that is called an arithmetic number. Try 4: its numbers 1, 2, and 4 add to 7, which can't be shared evenly by 3, so 4 isn't one.

Numbers With Whole-Number Divisor Averages

A divisor of a number is any whole number that divides it evenly. An arithmetic number is a number where the average of all its divisors is a whole number. To check, add up all the divisors and divide by how many there are. For 6, the divisors are 1, 2, 3, and 6; they add to 12, and there are 4 of them, so the average is 3, a whole number, and 6 is arithmetic. For 4, the divisors are 1, 2, and 4, adding to 7; 7 divided by 3 isn't whole, so 4 is not arithmetic.

Integer-Mean-of-Divisors Numbers

An arithmetic number is a positive integer n whose positive divisors have a whole-number average. Two standard functions describe the divisors: sigma(n), the sum of all positive divisors, and tau(n), how many divisors there are, each divisor counted once. The average of the divisors is sigma(n) divided by tau(n), so n is arithmetic exactly when tau(n) divides sigma(n) evenly. For example, 6 has divisors 1, 2, 3, 6, so sigma is 12 and tau is 4, and 12 divided by 4 is 3, so 6 is arithmetic. For 4, the divisors 1, 2, 4 give sigma 7 and tau 3, which doesn't divide evenly, so 4 is not.

 

An integer n is an arithmetic number when the arithmetic mean of its positive divisors is an integer, that is, when sigma(n), the sum of the positive divisors, is divisible by tau(n), the number of positive divisors. The construction maps the divisor multiset of n, with each positive divisor counted once, to two multiplicative-function statistics, and then applies a divisibility test to their ratio. Because both sigma and tau are multiplicative, each can be computed from the prime factorization of n, though the ratio condition itself is a divisibility property rather than a multiplicative one. For instance, 6 is arithmetic since sigma(6) = 12 and tau(6) = 4, whereas 4 is not since sigma(4) = 7 and tau(4) = 3. The defining conditions are precise: positive divisors are counted once, the mean equals sigma(n)/tau(n), and the quotient is an integer; changing any of these, such as including only proper divisors, defines a different set.

Scope of Application

Arithmetic number belongs to number theory and is useful where the analyst can specify the typed number theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate positive divisors are counted once, the arithmetic mean equals sigma(n) over tau(n), and the quotient is an integer. The scope is broad within that domain but bounded by the need for positive divisors are counted once, the arithmetic mean equals sigma(n) over tau(n), and the quotient is an integer. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.

Clarity

The abstraction clarifies a crowded vocabulary by making positive divisors are counted once, the arithmetic mean equals sigma(n) over tau(n), and the quotient is an integer the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Arithmetic number can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.

Manages Complexity

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Arithmetic number. Arithmetic number compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning

1. Identify the carrier. State what the elements, states, objects, or observations are: the typed number theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem.
2. Lock the constitutive rule. Express positive divisors are counted once, the arithmetic mean equals sigma(n) over tau(n), and the quotient is an integer independently of one notation or implementation.

Knowledge Transfer

Knowledge transfers strongly among subfields of number theory because they reuse the typed number theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, The divisor multiset maps n to two multiplicative-function statistics, and a divisibility test determines whether its average closes in the integers., and type the carrier, state every parameter and convention in the definition, test that positive divisors are counted once, the arithmetic mean equals sigma(n) over tau(n), and the quotient is an integer, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions

Current abstraction Arithmetic number Domain-specific

Parents (1) — outgoing typed relations

• Arithmetic number is a kind of Classification Prime

  What makes it its own entry: the domain-specific identity determined by positive divisors are counted once, the arithmetic mean equals sigma(n) over tau(n), and the quotient is an integer.

Hierarchy path (1) — routes to 1 parentless root

• Arithmetic number → is a kind of Classification

Neighborhood in Abstraction Space

Arithmetic number sits in a crowded region of the domain-specific corpus (1^st percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Integer & Divisor Number Theory (45 abstractions)

Nearest neighbors

• Arithmetic function — 0.95
• Highly composite number — 0.95
• Friendly number — 0.95
• Multiply perfect number — 0.94
• Multiplicative partition — 0.94

Computed from structural-signature embeddings · 2026-10-25