Multiply perfect number¶

A positive integer whose sum of positive divisors is an integer multiple k of the number itself.

Version: v1 · 2026-09-08 · History
Domain-specific #: 5703
Origin domain: number theory
Subdomain: number theory

Core Idea¶

A k-perfect or multiply perfect number n satisfies σ(n)=kn, generalizing perfect numbers at k=2 and linking prime factorization to multiplicative divisor-sum constraints. The divisor-sum function factors multiplicatively across coprime prime powers; their geometric sums must combine to exactly k times the original factorization. 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 integer n, divisor convention, divisor-sum function, and integer multiplier k satisfy the exact equation σ(n)=kn.

Scope of Application¶

Multiply perfect 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 integer n, divisor convention, divisor-sum function, and integer multiplier k satisfy the exact equation σ(n)=kn. The scope is broad within that domain but bounded by the need for positive integer n, divisor convention, divisor-sum function, and integer multiplier k satisfy the exact equation σ(n)=kn. 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 integer n, divisor convention, divisor-sum function, and integer multiplier k satisfy the exact equation σ(n)=kn 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 Multiply perfect 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 Multiply perfect number. Multiply perfect 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 integer n, divisor convention, divisor-sum function, and integer multiplier k satisfy the exact equation σ(n)=kn 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-sum function factors multiplicatively across coprime prime powers; their geometric sums must combine to exactly k times the original factorization., and type the carrier, state every parameter and convention in the definition, test that positive integer n, divisor convention, divisor-sum function, and integer multiplier k satisfy the exact equation σ(n)=kn, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Multiply perfect number Domain-specific

Parents (1) — more general patterns this builds on

Multiply perfect number is a kind of Ratio Prime

The proposed strict upward parent is prime:ratio.

Hierarchy path (1) — routes to 1 parentless root

Multiply perfect number → Ratio → Comparison → Self Checking

Neighborhood in Abstraction Space¶

Multiply perfect 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 — Number-Theoretic Sequences & Classes (37 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08