sociable number
Let an agent explain sociable numbers and aliquot cycles, relate them to perfect and amicable numbers, describe known cycles and open problems, and support computation and learning.
Bundle → Layer → Finding → Questions Filled
4 bundles · 8 layers · 8 findings · 16 questions
Understand What sociable numbers are.
Definition
Definition and aliquot sums.
Definition
Definition.
- What is the aliquot sum, and how do sociable numbers form cycles that generalise perfect and amicable numbers? definition
- Which entry fits perfect or amicable numbers? action
Examples
Known cycles.
Examples
Examples.
- Which sociable cycles are known, including the cycle beginning with 12496 and the 28-cycle, and how were they found? provenance
- Which databases list them? provenance
Compute Computing cycles.
Calculate
Aliquot sequences.
Calculate
Computing.
- How is the aliquot sequence of this number computed, and does it cycle, terminate or grow? action
- Which entry fits divisor functions? action
Search
Searching for cycles.
Search
Searching.
- How are sociable cycles searched for computationally, and what are the limits? provenance
- Which references are standard? provenance
Open Open problems.
Problems
Open questions.
Problems
Open questions.
- What is unknown about sociable numbers, such as cycles of length three, odd cycles and the Catalan-Dickson conjecture? boundary
- Which entry fits aliquot sequences? action
Theory
Theoretical results.
Theory
Theory.
- What theoretical results on the density of sociable numbers are known? provenance
- Which findings are recent? provenance
Learn History and teaching.
History
History.
History
History.
- How did the study of sociable numbers develop from Poulet in 1918 to modern computation? provenance
- Which entry fits history of number theory? action
Teach
Teaching.
Teach
Teaching.
- How can sociable numbers be used to teach divisors and computation? 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 belonging to a cycle of numbers in which each is the sum of the proper divisors of the previous one, returning to the start after a fixed number of steps, generalising perfect numbers with cycle length one and amicable pairs with cycle length two to cycles of length three or more, such as the cycle of five numbers beginning with 12496; sociable numbers are studied in number theory, with cycles found by computation and open questions about their existence and distribution.
Why it exists Filled
Let an agent explain sociable numbers and aliquot cycles, relate them to perfect and amicable numbers, describe known cycles and open problems, and support computation and learning.
Distinguishing features Filled
- Aliquot cycle
- Cycle length
- Computationally found
- Open problems
What robots and AI may and may not do Filled
Must not
- Claim a new sociable cycle without verification that others can repeat.
- Present open questions, such as whether cycles of length three exist, as settled.
- Attribute a discovery to a person without evidence.
- Confuse sociable numbers with amicable or perfect numbers.
Only with a human decision
- Announcing a claimed new discovery publicly.
May
- State known sociable cycles and their lengths with a reference.
- Search for or verify aliquot cycles computationally, reporting the search range.
- Explain how sociable numbers relate to perfect and amicable numbers.
Moral aspects Filled
- Mathematical claims depend on credit given to the right discoverers.
- False announcements waste others' effort and erode trust.
Who is affected
- Mathematicians and amateur researchers
- Students
Owners Filled
Steward
Nobody: a mathematical concept held in common; integer sequence databases record known cycles.
Master systems
- Integer sequence databases
Links to other meta-models Filled
parent
- Q16317911 - registry parent class
related
- positive integer - category
- Catalan number - another integer sequence entry
- Fermat number - another number theory entry
- cyclic process - cycles in general
What else AI and robots need to interact with it Filled
Identity and identifiers required Filled
- Vercy registry: vr.tr.sociable-number
- Wikidata: Q1149466 (https://www.wikidata.org/wiki/Q1149466)
Direct properties not applicable Not applicable
Not applicable
Plane XCT: no invented physical properties.
Recognition optional Filled
- Aliquot sum cycle of length three or more
- Generalises perfect and amicable numbers
- Perfect numbers have cycle length one; amicable pairs length two
- Not physical; sets of integers.
Capabilities and actions required Filled
- explain aliquot cycles
- relate to perfect and amicable numbers
- describe known cycles
- support computation
Hazards and failure modes required Filled
- Confusing sociable with amicable numbers
- Computational errors in divisor sums
- Overstating what is proven
Standards and interfaces required Filled
- Mathematical conventions and notation
- Sequence database conventions
Context of use required Filled
- Numbers in aliquot cycles.
- sociable numbers of order four
- sociable numbers of order five
- longer cycles
- related perfect and amicable numbers
- aliquot sequences generally
Sources Filled
- Wikidata item Q1149466: sociable number - identity and sense of the item
- Wikipedia: Sociable number - general description of the item
Open questions
- Should aliquot sequences be a separate entry?
- How should databases of cycles be linked?
- How should open problems be documented?
Machine files
Provenance
thing registry research (pass 2) · unreviewed
Built from: models/things/publications/thing-q1149466/spec.json