Quasiperfect number¶

Classify a positive integer by the divisor-sum equation sigma of n equals twice n plus one, an unresolved class whose hypothetical members must satisfy strong odd-square and prime-factor restrictions.

Version: v2 · 2026-08-30 · History
Domain-specific #: 2605
Origin domain: number theory
Subdomain: divisor sum number classes

Core Idea¶

A quasiperfect number is a positive integer \(n\) satisfying \(\sigma(n)=2n+1\), equivalently having proper-divisor sum \(\sigma(n)-n=n+1\).^[1] The divisor-sum equation fixes excess one above perfection, multiplicativity of the divisor-sum function converts prime factorization into strong congruence and size restrictions, and those restrictions narrow a class for which no member is currently known.

Its autonomous residual is the exact excess-one divisor-sum class and its unresolved existence status, not every abundant odd square, a perfect or almost-perfect number, or a heuristic near equality. The identity fails when proper and all-divisor sums are confused, one is omitted from the proper-divisor convention, numerical searches are called proofs of nonexistence, a necessary restriction is declared sufficient, or approximate arithmetic replaces equality.

Recognition requires an analyst to compute the divisor sum from a proven factorization, compare it exactly with twice the integer plus one, distinguish necessary from sufficient conditions, and preserve the difference between no known examples and a proof of nonexistence. Once established, it supports organizing divisor-sum problems around minimal abundance, deriving restrictions on hypothetical examples, comparing near-perfect number classes, and stating a long-standing open existence problem precisely without turning those uses into the definition.

Structural Signature¶

Carrier: positive integers equipped with their positive divisors, proper divisors, prime factorization, and the sum-of-divisors function
Inputs or antecedent state: integer, complete divisor set or factorization, divisor-sum value, aliquot sum, parity, square status, magnitude bound, number of distinct prime factors, and proof status
Constitutive operation: The divisor-sum equation fixes excess one above perfection, multiplicativity of the divisor-sum function converts prime factorization into strong congruence and size restrictions, and those restrictions narrow a class for which no member is currently known
Invariant: the positive integer satisfies the exact equality \(\sigma(n)=2n+1\); necessary conditions such as oddness, square form, lower bounds, and many prime factors do not replace that defining equation
Recognition test: compute the divisor sum from a proven factorization, compare it exactly with twice the integer plus one, distinguish necessary from sufficient conditions, and preserve the difference between no known examples and a proof of nonexistence
Output or consequence: organizing divisor-sum problems around minimal abundance, deriving restrictions on hypothetical examples, comparing near-perfect number classes, and stating a long-standing open existence problem precisely
Failure boundary: proper and all-divisor sums are confused, one is omitted from the proper-divisor convention, numerical searches are called proofs of nonexistence, a necessary restriction is declared sufficient, or approximate arithmetic replaces equality

What It Is Not¶

It is not the whole field of number theory; many objects in that field do not satisfy its constitutive rule.
It is not its canonical example. There is no known concrete quasiperfect integer, so the defining equation itself—not a displayed member—is the canonical example frame. That is an instance, not a definition.
It is not Hyperperfect Number. Hyperperfect numbers satisfy the parameterized relation \(n=1+k(\sigma(n)-n-1)\). Quasiperfect numbers instead have aliquot sum \(n+1\), so neither label is a generic synonym for near-perfect behavior.
It is not an unrestricted metaphor. Because no member is known, an encyclopedia entry must use conditional language for member properties and distinguish a search bound from a theorem applying to all hypothetical solutions

Scope of Application¶

Quasiperfect number applies when the analyst can specify positive integers equipped with their positive divisors, proper divisors, prime factorization, and the sum-of-divisors function and establish that the positive integer satisfies the exact equality \(\sigma(n)=2n+1\); necessary conditions such as oddness, square form, lower bounds, and many prime factors do not replace that defining equation. The entry concerns positive integers and the ordinary positive-divisor sum. Generalized divisor functions, polynomial analogues, and alternative quasiperfect terminology require separate definitions.^[2]

Recognition. compute the divisor sum from a proven factorization, compare it exactly with twice the integer plus one, distinguish necessary from sufficient conditions, and preserve the difference between no known examples and a proof of nonexistence
Comparison. Compare legitimate instances through divisor convention, sigma value, aliquot sum, abundance, parity, square form, magnitude, distinct prime factors, exponent pattern, search range, theorem date, and existence status.
Boundary. Because no member is known, an encyclopedia entry must use conditional language for member properties and distinguish a search bound from a theorem applying to all hypothetical solutions
Use. Preserve every assumption when using the identity for organizing divisor-sum problems around minimal abundance, deriving restrictions on hypothetical examples, comparing near-perfect number classes, and stating a long-standing open existence problem precisely.

Clarity¶

A clear claim names the carrier, governing rule, assumptions, and recognition test. This matters because quasiperfect sounds approximate, but the equality is exact; phrases such as minimal abundance vary with whether abundance means an integer difference or a normalized ratio. The disciplined statement is that the object counts as Quasiperfect number exactly when the positive integer satisfies the exact equality \(\sigma(n)=2n+1\); necessary conditions such as oddness, square form, lower bounds, and many prime factors do not replace that defining equation

Identity and measurement remain separate. Exact integer arithmetic and proved factorization determine membership; floating-point ratios, finite searches, and probable-prime factors cannot certify the equation or resolve universal existence. Approximation or noisy evidence may weaken a classification without changing its definition.

Manages Complexity¶

The abstraction compresses quasiperfect candidates, conditional factorization patterns, computational exclusions, stronger lower bounds, related perfect and almost-perfect classes, and generalized divisor-sum equations into a stable carrier, rule, invariant, and failure boundary. It makes comparison tractable while retaining the variables that control validity.

Compression can hide assumptions. A responsible use therefore declares divisor convention, sigma value, aliquot sum, abundance, parity, square form, magnitude, distinct prime factors, exponent pattern, search range, theorem date, and existence status and returns to the full diagnostic whenever a convention or boundary case changes.

Abstract Reasoning¶

Type the carrier. Establish positive integers equipped with their positive divisors, proper divisors, prime factorization, and the sum-of-divisors function and reject examples from a different problem.
Lock the rule. Express that the positive integer satisfies the exact equality \(\sigma(n)=2n+1\); necessary conditions such as oddness, square form, lower bounds, and many prime factors do not replace that defining equation independently of one notation or implementation.
Derive carefully. Infer organizing divisor-sum problems around minimal abundance, deriving restrictions on hypothetical examples, comparing near-perfect number classes, and stating a long-standing open existence problem precisely only under the stated assumptions.
Stress-test. Contrast the legitimate boundary case—Because no member is known, an encyclopedia entry must use conditional language for member properties and distinguish a search bound from a theorem applying to all hypothetical solutions—with this counterexample: an odd square with seven distinct prime factors is not thereby quasiperfect unless its divisor sum equals \(2n+1\).

Knowledge Transfer¶

Transfer within number theory is strong when new cases preserve the same carrier, mechanism, and diagnostic. The move from There is no known concrete quasiperfect integer, so the defining equation itself—not a displayed member—is the canonical example frame. to Theorems show that any quasiperfect number would have to be an odd square exceeding a very large lower bound and containing several distinct prime factors. demonstrates that continuity.^[3]

Outside the domain, only the skeleton—define a class by an exact near-balance equation and use global arithmetic structure to constrain hypothetical members even before existence is settled—travels automatically. The terms sum-of-divisors function, proper divisor, aliquot sum, abundance, odd square, prime factor, perfect number, necessary condition, and open problem retain domain-specific meanings, so every role and inference must be revalidated.

Examples¶

Canonical¶

There is no known concrete quasiperfect integer, so the defining equation itself—not a displayed member—is the canonical example frame. Any claimed discovery must provide a verified factorization or exhaustive divisor-sum computation showing exact equality; satisfying known odd-square and size bounds would only make it a candidate. It is canonical because the carrier, rule, invariant, and consequence are all inspectable.^[1]

Mapped back: positive integers equipped with their positive divisors, proper divisors, prime factorization, and the sum-of-divisors function → The divisor-sum equation fixes excess one above perfection, multiplicativity of the divisor-sum function converts prime factorization into strong congruence and size restrictions, and those restrictions narrow a class for which no member is currently known → the positive integer satisfies the exact equality \(\sigma(n)=2n+1\); necessary conditions such as oddness, square form, lower bounds, and many prime factors do not replace that defining equation → organizing divisor-sum problems around minimal abundance, deriving restrictions on hypothetical examples, comparing near-perfect number classes, and stating a long-standing open existence problem precisely

Applied / In Practice¶

Theorems show that any quasiperfect number would have to be an odd square exceeding a very large lower bound and containing several distinct prime factors. These deductions demonstrate the leverage of the divisor equation while leaving logical room for existence; every stronger published bound must be attached to its date and hypotheses. It qualifies only after the same diagnostic and failure boundary are checked.^[2]

Mapped back: declared instance → recognition test → boundary check → qualified use

Structural Tensions¶

T1: Exact identity vs. practical recognition. The constitutive condition may be exact while evidence is indirect. Diagnostic: Can the reviewer state both the condition and the warrant?
T2: Canonical form vs. variants. quasiperfect candidates, conditional factorization patterns, computational exclusions, stronger lower bounds, related perfect and almost-perfect classes, and generalized divisor-sum equations can preserve or change the identity. Diagnostic: Which named role is invariant across the variants?
T3: Compression vs. hidden assumptions. The label is useful only while prerequisites remain visible. Diagnostic: Can each downstream inference be traced to a declared assumption?
T4: Autonomy vs. reduction. The candidate uses broader structures but claims the exact excess-one divisor-sum class and its unresolved existence status, not every abundant odd square, a perfect or almost-perfect number, or a heuristic near equality. Diagnostic: Does that residual still support independent recognition after the parent and neighbors are subtracted?

Structural–Framed Character¶

The entry is structurally mixed but domain-framed. Its portable skeleton is define a class by an exact near-balance equation and use global arithmetic structure to constrain hypothetical members even before existence is settled; its identity-bearing terms are sum-of-divisors function, proper divisor, aliquot sum, abundance, odd square, prime factor, perfect number, necessary condition, and open problem. Those terms determine admissible objects, evidence, and consequences inside number theory.

Structural Core vs. Domain Accent¶

The structural core is a carrier governed by The divisor-sum equation fixes excess one above perfection, multiplicativity of the divisor-sum function converts prime factorization into strong congruence and size restrictions, and those restrictions narrow a class for which no member is currently known and tested by compute the divisor sum from a proven factorization, compare it exactly with twice the integer plus one, distinguish necessary from sufficient conditions, and preserve the difference between no known examples and a proof of nonexistence. The domain accent is constitutive rather than decorative, so an analogy that preserves only the skeleton is not another instance of Quasiperfect number.

The proposed strict upward parent is prime:constraint. The number class is literally cut out by one exact arithmetic equality on the divisor-sum function; the open existence question and derived factor restrictions supply the specialization. The edge is proposal-only and points to a frozen prior-baseline Prime.

The entry does not collapse into the parent because the exact excess-one divisor-sum class and its unresolved existence status, not every abundant odd square, a perfect or almost-perfect number, or a heuristic near equality A thematic neighbor is declined whenever it does not literally subsume that rule.

The prospective workspace queue contains one strict upward edge to prime:constraint. No live DAG mutation is authorized.

Relationships to Other Abstractions¶

Current abstraction Quasiperfect number Domain-specific

Parents (1) — more general patterns this builds on

Quasiperfect number is a kind of Constraint Prime

The proposed strict upward parent is prime:constraint.

Hierarchy path (1) — routes to 1 parentless root

Quasiperfect number → Constraint

Neighborhood in Abstraction Space¶

Quasiperfect number sits in a crowded region of the domain-specific corpus (13^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Integer Functions & Special Numbers (13 abstractions)

Nearest neighbors

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

Not to Be Confused With¶

Perfect number. Satisfies \(\sigma(n)=2n\), so its proper divisors sum exactly to \(n\).
Almost perfect number. Commonly satisfies \(\sigma(n)=2n-1\), one below the perfect all-divisor sum.
Multiply perfect number. Satisfies \(\sigma(n)=kn\) for an integer multiplier.
Almost-prime or probable candidate. Describes factor-count or evidential status and does not impose the quasiperfect divisor equation.

References¶

[1] E. Brown, H. Abbott, C. Aull, and D. Suryanarayana, 'Quasiperfect Numbers,' Acta Arithmetica 22(4), 439–447 (1973), DOI 10.4064/aa-22-4-439-447. registry ↩a ↩b

[2] Peter Hagis Jr. and Graeme L. Cohen, 'Some Results Concerning Quasiperfect Numbers,' Journal of the Australian Mathematical Society Series A 33(2), 275–286 (1982), DOI 10.1017/S1446788700018401. registry ↩a ↩b

[3] Richard K. Guy, Unsolved Problems in Number Theory, 3^rd ed., Springer, 2004, section B2, ISBN 978-0-387-20860-2. registry ↩