Refactorable number¶
Classify a positive integer as refactorable when the number of its positive divisors divides the integer itself.
Core Idea¶
A refactorable number, or tau number, is a positive integer \(n\) satisfying \(\tau(n)\mid n\), where \(\tau(n)\) counts the positive divisors of \(n\). If \(n=\prod_i p_i^{e_i}\), then \(\tau(n)=\prod_i(e_i+1)\). The defining test first computes divisor count from the prime-exponent vector and then tests divisibility of \(n\) by that count. The property defines an infinite but sparse integer sequence.
Prime factorization supplies divisor count without enumerating each divisor. The exponent product yields \(\tau(n)\); modular division then decides membership. For \(18=2\cdot3^2\), the count is \((1+1)(2+1)=6\), and \(6\mid18\), so 18 is refactorable.
Scope of Application¶
The abstraction is literal wherever practitioners can identify the same constitutive roles, apply the same boundary tests, and obtain the same kind of output. The following habitats are uses of Refactorable number itself, not metaphors based only on resemblance.
- Elementary number theory. Studying interaction between \(n\) and its divisor count.
- Integer-sequence research. Enumerating values and testing conjectured patterns.
- Analytic number theory. Estimating counting functions and natural density.
- Congruence analysis. Constraining refactorable numbers in residue classes.
- Machine discovery. Examining rediscovery and automated concept formation.
- Algorithmic enumeration. Generating candidates from factorizations with correctness checks.
Clarity¶
A clear account of Refactorable number must preserve the recognition invariant stated in the Core Idea rather than rely on the title alone. Define \(\tau(n)\) as the count of positive divisors and fix the domain \(n\ge1\). Use the prime-exponent formula under a verified factorization. Separate infinitude, natural density, and counting-function asymptotics. Distinguish historical priority from the later refactorable name. These declarations are not editorial extras: each changes what observations count, which transformations are licensed, and what conclusion can be drawn.
Manages Complexity¶
Refactorable number manages complexity by replacing a diffuse field of observations or possible operations with a bounded role structure: positive integer supplies a number \(n\ge1\) supplies the candidate.; prime factorization supplies exponents determine the divisor count.; divisor-count function supplies the arithmetic function \(\tau(n)\) supplies the internal statistic.; divisibility test supplies membership requires zero remainder on division by \(\tau(n)\).; integer sequence supplies all satisfying values form an ordered sparse family..
Abstract Reasoning¶
- Factor the positive integer into prime powers. 2. Multiply each exponent plus one to obtain \(\tau(n)\). 3. Divide \(n\) by \(\tau(n)\) and inspect the remainder. 4. Classify membership without importing unrelated divisor-sum properties. 5. For sequence claims, verify the generation range and duplicate handling. 6. State whether a theorem proves infinitude, density, or only a bound. 7. Test conjectures against known congruence and adjacency results.
Knowledge Transfer¶
The strict upward abstraction is Predicate. Refactorable Number instantiates Predicate because \(\tau(n)\mid n\) assigns a definite truth value to each positive integer. Within divisor function predicate, the full mechanism transfers literally when the same roles and boundary tests recur. Beyond that domain, only the parent-level skeleton should travel. Reusing the label Refactorable number after removing its constitutive vocabulary would hide a change of mechanism behind an analogy. The honest transfer rule is therefore two-stage: recognize the domain-specific pattern first, then lift only the parent relation that remains invariant under a substrate change.
Relationships to Other Abstractions¶
Current abstraction Refactorable number Domain-specific
Parents (1) — more general patterns this builds on
-
Refactorable number is a kind of Predicate Prime
Refactorable Number instantiates Predicate because \(\tau(n)\mid n\) assigns a definite truth value to each positive integer.
Neighborhood in Abstraction Space¶
Refactorable number sits in a sparse region of the domain-specific corpus (76th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Computational Number Theory & Enumeration (13 abstractions)
Nearest neighbors
- Sublime number — 0.84
- Fermat number — 0.84
- Divisor Function — 0.83
- Arithmetic number — 0.82
- Smooth Number — 0.82
Computed from structural-signature embeddings · 2026-09-08