highly abundant number
Let an agent define highly abundant numbers, relay their sequence and known properties from number theory references, and distinguish them from abundant, superabundant and highly composite numbers.
Bundle → Layer → Finding → Questions Filled
4 bundles · 8 layers · 8 findings · 16 questions
Define Definition.
Definition
Definition.
Definition
Definition.
- What is a highly abundant number, and how is it checked? definition
- Is the question about highly abundant, abundant, superabundant or highly composite numbers? boundary
Sequence
Sequence.
Sequence
Sequence.
- What are the first highly abundant numbers, and where is the sequence catalogued? measurement
- Which entry fits divisor function? action
Results Results.
Properties
Known properties.
Properties
Properties.
- What did Alaoglu and Erdos prove about highly abundant numbers? provenance
- Which references are standard? provenance
Relations
Relations to other classes.
Relations
Relations.
- How do highly abundant numbers relate to superabundant and highly composite numbers? provenance
- Which entry fits superabundant number? action
Compute Computation.
Algorithms
Algorithms.
Algorithms
Algorithms.
- How are highly abundant numbers computed? provenance
- Which sources are cited? provenance
Open
Open questions.
Open
Open.
- What open questions exist about odd members and asymptotics? provenance
- Which entry fits open problems in number theory? action
Context History.
History
History.
History
History.
- How did Pillai, Alaoglu and Erdos introduce and study these numbers? provenance
- Which entry fits the history of number theory? action
Related
Record-setting classes.
Related
Related.
- What other record-setting integer classes exist, from Ramanujan highly composite numbers onward? provenance
- Which entry fits highly composite number? 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 n whose sum of divisors, including n itself, is greater than the sum of divisors of every smaller positive integer, the sequence beginning 1, 2, 3, 4, 6, 8, 10, 12, 16, 18, 20, 24, 30, 36, 42, 48, 60; highly abundant numbers were introduced by Pillai in 1943 and studied by Alaoglu and Erdos, are related to superabundant and highly composite numbers, and are catalogued in the OEIS.
Why it exists Filled
Let an agent define highly abundant numbers, relay their sequence and known properties from number theory references, and distinguish them from abundant, superabundant and highly composite numbers.
Distinguishing features Filled
- Record divisor sum
- Introduced by Pillai
- Related to superabundant numbers
- Not all even
What robots and AI may and may not do Filled
Must not
- Confuse highly abundant numbers with abundant, superabundant or highly composite numbers.
- Attribute the concept to the wrong mathematician.
- Publish computed terms without the search bound and method.
- Present an unverified conjecture about the sequence as proven.
Only with a human decision
- Publishing a claimed new result about the sequence.
May
- State the defining property of highly abundant numbers and the start of the sequence.
- Compute divisor sums and test the property within stated limits.
Moral aspects Filled
- Similar-sounding number classes are easily mixed up, which misleads learners.
- Attribution of mathematical ideas matters to the record.
Who is affected
- Mathematicians
- Students
- Sequence database editors
Owners Filled
Steward
Nobody owns a mathematical fact; sequence databases record it by convention.
Links to other meta-models Filled
parent
- Q16317911 - registry parent class
related
- positive integer - in registry terms
- divisor function - sigma(n)
- superabundant number - a similar record class
- OEIS - as A002093
What else AI and robots need to interact with it Filled
Identity and identifiers required Filled
- Vercy registry: vr.tr.highly-abundant-number
- Wikidata: Q3205712 (https://www.wikidata.org/wiki/Q3205712)
- OEIS: A002093
Direct properties not applicable Not applicable
- smallest: 1 number
Plane XCT: no invented physical properties.
Recognition optional Filled
- Record-setting divisor sums
- Sequence OEIS A002093
- Abundant numbers exceed twice themselves; superabundant numbers set records for sigma(n)/n
- The sequence 1, 2, 3, 4, 6, 8, 10, 12, 16, 18, 20, 24, 30, ...
Capabilities and actions required Filled
- define and list
- relay known properties
- relate to similar classes
- distinguish from related concepts
Hazards and failure modes required Filled
- Confusing with abundant or highly composite numbers
- Assuming all superabundant numbers are highly abundant or vice versa
- Computation errors
Standards and interfaces required Filled
- No regulation; mathematical conventions
Context of use required Filled
- Not applicable; a class of integers.
- highly abundant numbers in general
- superabundant numbers as a subsequence in many cases
- highly composite numbers as a related record class
- colossally abundant numbers
Sources Filled
- Wikidata item Q3205712: highly abundant number - identity and sense of the item
- Wikipedia: Highly abundant number - general description of the item
Open questions
- What is the relation between highly abundant and superabundant numbers beyond known results?
- How should OEIS and literature be linked?
- Are there infinitely many odd highly abundant numbers?
Machine files
Provenance
thing registry research (pass 2) · unreviewed
Built from: models/things/publications/thing-q3205712/spec.json