practical number
Let an agent define practical numbers, relay their characterisation, properties and known results from number theory references, and distinguish them from related classes such as perfect and abundant numbers.
Bundle → Layer → Finding → Questions Filled
4 bundles · 8 layers · 8 findings · 16 questions
Define Definition.
Definition
Definition.
Definition
Definition.
- What is a practical number, and how is it checked? definition
- Is the question about practical, perfect or abundant numbers? boundary
Characterisation
Characterisation.
Characterisation
Characterisation.
- What is the Stewart-Sierpinski characterisation in terms of prime factorisation? definition
- Which entry fits prime factorisation? action
Results Results.
Properties
Properties.
Properties
Properties.
- What properties do practical numbers have, including closure under products? provenance
- Which references are standard? provenance
Primes
Analogies with primes.
Primes
Primes.
- What Goldbach-type and density results hold for practical numbers, as proved by Melfi, Weingartner and others? provenance
- Which sources are cited? provenance
Apply Applications.
Egyptian
Egyptian fractions.
Egyptian
Egyptian.
- How are practical numbers used in Egyptian fraction representations? provenance
- Which entry fits Egyptian fraction? action
Computation
Computation.
Computation
Computation.
- How are practical numbers computed and enumerated? provenance
- Which entry fits computational number theory? action
Context History.
History
History.
History
History.
- How did Srinivasan, Stewart and Sierpinski introduce and characterise practical numbers? provenance
- Which entry fits the history of number theory? action
Related
Related classes.
Related
Related.
- How do practical numbers relate to perfect, abundant and highly composite numbers? provenance
- Which entry fits perfect 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 such that every smaller positive integer can be written as a sum of distinct divisors of n, the sequence beginning 1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, named by Srinivasan in 1948, characterised by a condition on prime factorisation by Stewart and Sierpinski, and including all powers of two and all perfect numbers; practical numbers behave in some ways like primes, and Goldbach-type results and density estimates have been proved for them.
Why it exists Filled
Let an agent define practical numbers, relay their characterisation, properties and known results from number theory references, and distinguish them from related classes such as perfect and abundant numbers.
Distinguishing features Filled
- Subset sum property of divisors
- Stewart-Sierpinski characterisation
- Prime-like distribution
- All even except 1
What robots and AI may and may not do Filled
Must not
- Present a conjectured property as proven.
- Confuse practical numbers with similar sequences such as highly composite numbers.
- Credit the naming to someone other than Srinivasan.
Only with a human decision
- Publishing new results on practical numbers as established.
May
- Test whether an integer is practical and list its divisors as a witness.
- Explain properties of practical numbers and their link to Egyptian fractions.
Moral aspects Filled
- Accurate credit and proof standards keep mathematics trustworthy.
- Students rely on correct definitions.
Who is affected
- Mathematicians
- Students
Owners Filled
Steward
The public: held in common as mathematical knowledge.
Links to other meta-models Filled
parent
- Q16317911 - registry parent class
related
- positive integer - in registry terms
- prime factorization - via the Stewart-Sierpinski condition
- perfect number - as a subclass
- OEIS - as A005153
What else AI and robots need to interact with it Filled
Identity and identifiers required Filled
- Vercy registry: vr.tr.practical-number
- Wikidata: Q577211 (https://www.wikidata.org/wiki/Q577211)
- OEIS: A005153
Direct properties not applicable Not applicable
- smallest: 1 number
- smallest odd practical number: 1 number - the only odd one
Plane XCT: no invented physical properties.
Recognition optional Filled
- Integers whose divisors sum to all smaller integers
- Sequence OEIS A005153
- Perfect numbers equal their aliquot sum; abundant numbers exceed it
- The sequence 1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, ...
Capabilities and actions required Filled
- define and characterise
- relay properties and results
- relay analogies with primes
- distinguish related classes
Hazards and failure modes required Filled
- Confusing with perfect or abundant numbers
- Misstating the characterisation
- Computation errors
Standards and interfaces required Filled
- No regulation; mathematical conventions
Context of use required Filled
- Not applicable; a class of integers.
- practical numbers in general
- powers of two and perfect numbers as practical numbers
- primitive practical numbers
- practical numbers in Egyptian fraction problems
Sources Filled
- Wikidata item Q577211: practical number - identity and sense of the item
- Wikipedia: Practical number - general description of the item
Open questions
- How should the density and Goldbach-type results be linked?
- How should OEIS and literature be linked?
- How should primitive practical numbers be recorded?
Machine files
Provenance
thing registry research (pass 2) · unreviewed
Built from: models/things/publications/thing-q577211/spec.json