Arithmetic number¶
A positive integer whose positive divisors have an integer arithmetic mean.
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.
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¶
- 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) — more general patterns this builds on
-
Arithmetic number is a kind of Classification Prime
The proposed strict upward parent is
prime:classification.
Hierarchy path (1) — routes to 1 parentless root
- Arithmetic number → Classification
Neighborhood in Abstraction Space¶
Arithmetic number sits in a crowded region of the domain-specific corpus (2nd percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Number-Theoretic Sequences & Classes (37 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 · 2026-09-08