friendly number
Let an agent explain friendly numbers and abundancy, relay known results and open problems from number theory references, and distinguish friendly numbers from amicable, perfect and sociable numbers.
Bundle → Layer → Finding → Questions Filled
4 bundles · 8 layers · 8 findings · 16 questions
Understand What a friendly number is.
Definition
Definition.
Definition
Definition.
- What is a friendly number, what is abundancy, and what is a solitary number? definition
- Is the question about friendly numbers, amicable numbers or perfect numbers? boundary
Examples
Examples.
Examples
Examples.
- What are examples of friendly pairs and clubs, and which small numbers are known to be solitary? measurement
- Which entry fits the specific number class? action
Theory Results.
Results
Known results.
Results
Results.
- What is proved about solitary numbers, such as those coprime to their divisor sum? provenance
- Which references are standard? provenance
Open
Open problems.
Open
Open.
- Why is the status of 10 unknown, and what would resolve it? provenance
- Which sources are cited? provenance
Compare Related concepts.
Amicable
Amicable and sociable numbers.
Amicable
Amicable.
- How do amicable, sociable and perfect numbers relate to friendly numbers? provenance
- Which entry fits amicable number? action
Abundancy
Abundancy index.
Abundancy
Abundancy.
- What properties does the abundancy index have, and how does it relate to abundant and deficient numbers? provenance
- Which entry fits abundant number? action
Context History and computation.
History
History.
History
History.
- How did Anderson and Hickerson introduce friendly numbers, and how has the study developed? provenance
- Which entry fits the history of number theory? action
Computation
Computation.
Computation
Computation.
- How can friendly numbers be searched for computationally? action
- Which entry fits computational number theory? 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 shares its abundancy index, the ratio of the sum of its divisors to itself, with at least one other integer, so that the two form a friendly pair or club, as 6 and 28 with abundancy 2, while an integer with no such partner is solitary, as are all primes and prime powers; whether numbers such as 10 are friendly or solitary is unknown, and the concept is distinct from amicable numbers.
Why it exists Filled
Let an agent explain friendly numbers and abundancy, relay known results and open problems from number theory references, and distinguish friendly numbers from amicable, perfect and sociable numbers.
Distinguishing features Filled
- Abundancy index
- Pairs and clubs
- Solitary numbers
- Open cases
What robots and AI may and may not do Filled
Must not
- Claim that unresolved numbers are solitary or friendly without proof.
- Confuse friendly numbers with amicable numbers.
- Present computational searches as proofs.
- Invent results or credit them wrongly.
Only with a human decision
- Publishing a claimed resolution of an open case.
May
- Explain friendly numbers and the abundancy index with examples.
- Check whether given numbers form a friendly pair.
- State which numbers, such as 10, are unresolved.
Moral aspects Filled
- Mathematical honesty requires separating what is proved from what is believed.
- Credit for results matters to researchers.
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
- abundancy index - sigma(n)/n
- amicable number - a different divisor relation
- perfect number - as friendly numbers of abundancy 2
What else AI and robots need to interact with it Filled
Identity and identifiers required Filled
- Vercy registry: vr.tr.friendly-number
- Wikidata: Q1707307 (https://www.wikidata.org/wiki/Q1707307)
- OEIS: A074902 friendly numbers
- OEIS: A095739 known solitary numbers
Direct properties not applicable Not applicable
- abundancy of 6 and 28: 2 value - all perfect numbers are friendly
- smallest unresolved number: 10 value - unknown whether friendly or solitary
- concept introduced: 1977 year - Anderson and Hickerson
Plane XCT: no invented physical properties.
Recognition optional Filled
- Integers sharing an abundancy index
- Friendly pairs and clubs; solitary numbers
- Amicable numbers have divisor sums equal to each other; perfect numbers have abundancy exactly 2
- Not a visible object; integers grouped by abundancy.
Capabilities and actions required Filled
- explain the definition
- relay known results
- describe open problems
- distinguish from related numbers
Hazards and failure modes required Filled
- Confusing friendly with amicable numbers
- Assuming unresolved numbers are solitary
- Misreading abundancy as a ratio of sums
Standards and interfaces required Filled
- No regulation; standard mathematical definitions
Context of use required Filled
- Not applicable; a mathematical concept.
- friendly pairs
- friendly clubs with three or more members
- solitary numbers including primes and prime powers
- perfect numbers as a friendly club with abundancy 2
- unresolved numbers such as 10, 14, 15, 20
Sources Filled
- Wikidata item Q1707307: friendly number - identity and sense of the item
- Wikipedia: Friendly number - general description of the item
Open questions
- Should abundancy index be a separate entry?
- How should number theory references be linked?
- How should the status of unresolved numbers be tracked?
Machine files
Provenance
thing registry research (pass 2) · unreviewed
Built from: models/things/publications/thing-q1707307/spec.json