refactorable number
Let an agent explain refactorable numbers, relay their definition, examples and known properties from number theory references, and distinguish them from related divisor-based number classes.
Bundle → Layer → Finding → Questions Filled
4 bundles · 8 layers · 8 findings · 16 questions
Understand What a refactorable number is.
Definition
Definition.
Definition
Definition.
- What is a refactorable number, and why is 12 refactorable while 10 is not? definition
- Is the question about refactorable numbers or another divisor-based class? boundary
Examples
Examples.
Examples
Examples.
- What are the first refactorable numbers, and which perfect squares and prime powers qualify? measurement
- Which entry fits the specific number? action
Theory Results.
Results
Known results.
Results
Results.
- What is proved about density and consecutive refactorable numbers, and by whom? provenance
- Which references are standard? provenance
Open
Open questions.
Open
Open.
- What questions about refactorable numbers remain open? provenance
- Which sources are cited? provenance
Compare Related classes.
Divisor
Divisor function.
Divisor
Divisor.
- How does the divisor counting function tau behave, and how does it determine refactorability? provenance
- Which entry fits divisor function? action
Classes
Other classes.
Classes
Classes.
- How do refactorable numbers relate to highly composite, perfect and harmonic divisor numbers? provenance
- Which entry fits highly composite number? action
Context History and computation.
History
History.
History
History.
- How were refactorable numbers introduced by Kennedy and Cooper and named tau numbers by Colton? provenance
- Which entry fits the history of number theory? action
Computation
Computation.
Computation
Computation.
- How can refactorable numbers be computed programmatically? action
- Which entry fits integer sequence? action
Classifiers Filled
- Family
- Thing Registry
- Category
- Cross-cutting context
- Entry kind
- thing
- Plane
- XCT
- Domain
- XCT.QTY
What it is Filled
A positive integer that is divisible by the number of its divisors, also called a tau number, such as 1, 2, 8, 9, 12, 18, 24, 36, 40 and 56, so that 12 with six divisors is refactorable because 6 divides 12; refactorable numbers have natural density zero, can occur in consecutive pairs such as 8 and 9, and were studied by Kennedy and Cooper and by Zelinsky.
Why it exists Filled
Let an agent explain refactorable numbers, relay their definition, examples and known properties from number theory references, and distinguish them from related divisor-based number classes.
Distinguishing features Filled
- The divisor count divides the number.
- Refactorable numbers have natural density zero.
- Consecutive refactorable numbers include 8 and 9.
- The definition uses the divisor-counting tau function.
What robots and AI may and may not do Filled
Must not
- Miscount divisors or present wrong examples.
- Overstate open results as proved.
- Confuse tau numbers with other divisor classes.
- Misattribute results.
Only with a human decision
- Publishing a new claimed theorem about refactorable numbers.
May
- Explain refactorable numbers and test given integers.
- Count divisors correctly and show the working.
- Cite known results with their authors.
Moral aspects Filled
- Correct credit for results matters to mathematicians.
- Errors in teaching material mislead students.
Who is affected
- Students
- Researchers
- Readers
Owners Filled
Steward
Nobody owns the concept; the mathematical community maintains it.
Links to other meta-models Filled
parent
- Q16317911 - registry parent class
related
- positive integer - in registry terms
- divisor function - tau(n)
- OEIS - as A033950
- perfect number - defined by divisor sums
What else AI and robots need to interact with it Filled
Identity and identifiers required Filled
- Vercy registry: vr.tr.refactorable-number
- Wikidata: Q2063121 (https://www.wikidata.org/wiki/Q2063121)
- OEIS: A033950 refactorable numbers
Direct properties not applicable Not applicable
- first terms: 1, 2, 8, 9, 12, 18, 24, 36, 40, 56 sequence - OEIS A033950
- natural density: 0 value - proved by Kennedy and Cooper
- consecutive refactorable numbers: only 1 and 2 result - Zelinsky
Plane XCT: no invented physical properties.
Recognition optional Filled
- Integers divisible by their number of divisors
- Tau numbers 1, 2, 8, 9, 12, 18, 24, 36, 40, 56
- Perfect numbers use divisor sums; highly composite numbers maximise divisor counts
- Not a visible object; integers with a divisibility property.
Capabilities and actions required Filled
- explain the definition
- relay examples and properties
- describe results
- distinguish related classes
Hazards and failure modes required Filled
- Confusing tau numbers with other divisor classes
- Miscounting divisors
- Overstating open results
Standards and interfaces required Filled
- No regulation; standard mathematical definitions
Context of use required Filled
- Not applicable; a mathematical concept.
- refactorable numbers
- refactorable numbers that are perfect squares
- refactorable primes powers such as 8 and 9
- related classes such as highly composite and tau-rich numbers
Sources Filled
- Wikidata item Q2063121: refactorable number - identity and sense of the item
- Wikipedia: Refactorable number - general description of the item
Open questions
- Should the divisor function be the primary linked entry?
- How should number theory references be linked?
- How should open problems such as Colton conjectures be recorded?
Machine files
Provenance
thing registry research (pass 2) · unreviewed
Built from: models/things/publications/thing-q2063121/spec.json