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.
Core Idea¶
A quasiperfect number is a positive integer \(n\) satisfying \(\sigma(n)=2n+1\), equivalently having proper-divisor sum \(\sigma(n)-n=n+1\). 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.
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.
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
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. 2. 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.
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.
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 (13th 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
- Deficient number — 0.95
- Perfect number — 0.95
- Hyperperfect number — 0.95
- Sublime number — 0.93
- Multiply perfect number — 0.91
Computed from structural-signature embeddings · 2026-09-08