Published
harmonic divisor number
vr.tr.harmonic-divisor-number · thing-q60679
Let an agent define harmonic divisor numbers precisely, test a given integer, relate them to perfect numbers and open conjectures, and point to sequence references.
Thing Registry Cross-cutting context XCT.QTY
Bundle → Layer → Finding → QuestionsClassifiersWhat it isWhy it existsDistinguishing featuresWhat robots and AI may and may not doMoral aspectsOwnersLinks to other meta-modelsIdentity and identifiersDirect propertiesRecognitionCapabilities and actionsHazards and failure modesStandards and interfacesContext of useSources
Bundle → Layer → Finding → Questions Filled
4 bundles · 8 layers · 8 findings · 16 questions
Define The definition.
Definition
Definition.
Definition
Definition.
- What is the harmonic mean of the divisors, and when is it an integer? definition
- Is the question about harmonic divisor numbers or a related class? boundary
Examples
Examples.
Examples
Examples.
- What are the harmonic divisor numbers below a given bound? measurement
- Which entry fits integer sequences? action
Theory Properties.
Perfect
Perfect numbers.
Perfect
Perfect.
- Why is every perfect number a harmonic divisor number? provenance
- Which entry fits perfect numbers? action
Conjectures
Open problems.
Conjectures
Conjectures.
- What is known and conjectured about odd harmonic divisor numbers and their density? provenance
- What is proven versus conjectured? boundary
Compute Computation.
Test
Testing.
Test
Test.
- How is an integer tested using the number and sum of divisors? action
- What is the result for the number in question? measurement
Search
Searches.
Search
Search.
- How far have harmonic divisor numbers been enumerated, with attribution? provenance
- Which references are standard? provenance
Context History and teaching.
History
Ore.
History
History.
- How did Ore introduce these numbers, and how did later work develop them? provenance
- Which entry fits Oystein Ore? action
Teach
Teaching.
Teach
Teaching.
- How are harmonic divisor numbers used in teaching? action
- Which misconceptions arise? provenance
Classifiers Filled
- Family
- Thing Registry
- Category
- Cross-cutting context
- Entry kind
- thing
- Plane
- XCT
- Domain
- XCT.QTY
What it is Filled
A positive integer whose divisors have an integer harmonic mean, also called an Ore number after Oystein Ore who introduced them in 1948; the first are 1, 6, 28, 140, 270 and 496, every perfect number is a harmonic divisor number, and it is conjectured that no odd harmonic divisor number greater than 1 exists.
Why it exists Filled
Let an agent define harmonic divisor numbers precisely, test a given integer, relate them to perfect numbers and open conjectures, and point to sequence references.
Distinguishing features Filled
- Defined by the harmonic mean of divisors
- Contains all perfect numbers
- Conjectured to have no odd members beyond 1
- Named after Ore
What robots and AI may and may not do Filled
Must not
- Present the conjecture that no odd harmonic divisor number exceeds 1 as proven.
- Report a number as a harmonic divisor number without checking its harmonic mean exactly.
- Confuse harmonic divisor numbers with arithmetic or refactorable numbers.
- Attribute results to Oystein Ore or other mathematicians that they did not publish.
Only with a human decision
- Announcing a claimed proof or counterexample as a new result.
May
- Define harmonic divisor numbers and test whether a given integer is one.
- Relate them to perfect numbers and cite sequence references.
- Explain the conjecture about odd harmonic divisor numbers as still open.
Moral aspects Filled
- Accurate attribution respects the mathematicians who did the work.
- Overstated claims in number theory mislead students and waste other people's checking time.
Who is affected
- Students and learners
- Mathematicians
- Readers of references
Owners Filled
Steward
Nobody owns the concept; the mathematical community and sequence databases maintain the record.
Links to other meta-models Filled
parent
- Q16317911 - registry parent class
related
- positive integer - a property of integers
- perfect number - every perfect number qualifies
- On-Line Encyclopedia of Integer Sequences - sequence A001599
- harmonic mean - the defining average
What else AI and robots need to interact with it Filled
Identity and identifiers required Filled
- Vercy registry: vr.tr.harmonic-divisor-number
- Wikidata: Q60679 (https://www.wikidata.org/wiki/Q60679)
- OEIS: A001599 harmonic or Ore numbers
Direct properties not applicable Not applicable
- first terms: 1, 6, 28, 140, 270, 496, 672 integers
Plane XCT: no invented physical properties.
Recognition optional Filled
- Divisors have integer harmonic mean
- Examples 1, 6, 28, 140, 270, 496
- Perfect numbers are a subset; abundant numbers are a different notion
- Not visible; a property of integers.
Capabilities and actions required Filled
- test whether an integer is a harmonic divisor number
- list small examples
- explain the link to perfect numbers
- state open problems
Hazards and failure modes required Filled
- Confusing with arithmetic or refactorable numbers
- Presenting the odd conjecture as proven
- Errors in computing the harmonic mean
Standards and interfaces required Filled
- No regulation; definitions follow the literature and the OEIS
Context of use required Filled
- Not for anything; a curiosity in number theory.
- harmonic divisor numbers
- perfect numbers as a subset
- related notions such as refactorable and arithmetic numbers
Sources Filled
- Wikidata item Q60679: harmonic divisor number - identity and sense of the item
- Wikipedia: Harmonic divisor number - general description of the item
Open questions
- Should perfect numbers be linked as the primary related entry?
- How should OEIS entries be linked?
- How should the odd conjecture be recorded?
Machine files
Provenance
thing registry research (pass 2) · unreviewed
Built from: models/things/publications/thing-q60679/spec.json