superabundant number
Let an agent define superabundant numbers, relay their sequence and properties from number theory references, explain their link to the Riemann hypothesis, and distinguish them from highly abundant and colossally abundant numbers.
Bundle → Layer → Finding → Questions Filled
4 bundles · 8 layers · 8 findings · 16 questions
Define Definition.
Definition
Definition.
Definition
Definition.
- What is a superabundant number, and how is it checked? definition
- Is the question about superabundant, highly abundant or colossally abundant numbers? boundary
Sequence
Sequence.
Sequence
Sequence.
- What are the first superabundant numbers, and where is the sequence catalogued? measurement
- Which entry fits divisor function? action
Results Results.
Structure
Structure.
Structure
Structure.
- What did Alaoglu and Erdos prove about the prime factorisation of superabundant numbers? provenance
- Which references are standard? provenance
Riemann
Riemann hypothesis link.
Riemann
Riemann.
- How does Robin inequality connect superabundant and colossally abundant numbers to the Riemann hypothesis? provenance
- Which entry fits Robin theorem? action
Compute Computation.
Algorithms
Algorithms.
Algorithms
Algorithms.
- How are superabundant numbers computed, and how far have they been enumerated? provenance
- Which sources are cited? provenance
Related
Related classes.
Related
Related.
- How do superabundant numbers relate to highly composite and colossally abundant numbers? provenance
- Which entry fits colossally abundant number? action
Context History.
History
History.
History
History.
- How did Ramanujan, Alaoglu and Erdos develop the study of these numbers? provenance
- Which entry fits the history of number theory? action
Open
Open questions.
Open
Open.
- What open problems exist about superabundant numbers? provenance
- Which entry fits open problems in number theory? action
Classifiers Filled
- Family
- Thing Registry
- Category
- Cross-cutting context
- Entry kind
- thing
- Plane
- XCT
- Domain
- XCT.QTY
- Other names and narrower kinds
- colossally abundant number
What it is Filled
A positive integer n for which the ratio of the sum of its divisors to n itself is greater than that ratio for every smaller positive integer, the sequence beginning 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360; superabundant numbers were defined by Alaoglu and Erdos in 1944, are all highly abundant, are closely related to colossally abundant numbers, and connect to the Riemann hypothesis through Robin inequality.
Why it exists Filled
Let an agent define superabundant numbers, relay their sequence and properties from number theory references, explain their link to the Riemann hypothesis, and distinguish them from highly abundant and colossally abundant numbers.
Distinguishing features Filled
- Record abundancy ratio
- Prime factorisation structure
- Robin inequality link
- Subset of highly abundant numbers
What robots and AI may and may not do Filled
Must not
- Claim that a computation proves or disproves the Riemann hypothesis without peer review.
- Present unverified terms of the sequence as checked.
- Attribute results to mathematicians without sources.
Only with a human decision
- Publishing claims of a new mathematical result.
May
- Compute, list and verify superabundant numbers.
- Explain their relation to highly composite numbers and the Riemann hypothesis.
Moral aspects Filled
- Bold claims about famous open problems can mislead students and the public.
- Mathematicians' credit for results deserves accurate attribution.
Who is affected
- Students
- Mathematicians
- Readers of popular mathematics
Owners Filled
Steward
Nobody owns a number sequence; it belongs to shared mathematical knowledge.
Master systems
- Integer sequence databases
Links to other meta-models Filled
parent
- Q3205712 - registry parent class
related
- highly abundant number - in registry terms
- divisor function - sigma(n)/n
- Riemann hypothesis - through Robin inequality
- OEIS - as A004394
What else AI and robots need to interact with it Filled
Identity and identifiers required Filled
- Vercy registry: vr.tr.superabundant-number
- Wikidata: Q3280976 (https://www.wikidata.org/wiki/Q3280976)
- OEIS: A004394
Direct properties not applicable Not applicable
- smallest: 1 number
Plane XCT: no invented physical properties.
Recognition optional Filled
- Record values of sigma(n)/n
- Sequence OEIS A004394
- Highly abundant numbers set records for sigma(n); colossally abundant numbers use a stricter criterion
- The sequence 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, ...
Capabilities and actions required Filled
- define and list
- relay properties
- explain the Riemann link
- distinguish related classes
Hazards and failure modes required Filled
- Confusing with highly abundant or highly composite numbers
- Overstating the Riemann hypothesis connection
- Computation errors
Standards and interfaces required Filled
- No regulation; mathematical conventions
Context of use required Filled
- Not applicable; a class of integers.
- superabundant numbers in general
- colossally abundant numbers as a subsequence
- highly abundant numbers as the containing class
- highly composite numbers as a related class
Sources Filled
- Wikidata item Q3280976: superabundant number - identity and sense of the item
- Wikipedia: Superabundant number - general description of the item
Open questions
- Are all superabundant numbers beyond some point colossally abundant or not?
- How should OEIS and literature be linked?
- How should the Riemann link be presented accurately?
Machine files
Provenance
thing registry research (pass 2) · unreviewed
Built from: models/things/publications/thing-q3280976/spec.json