untouchable number
Let an agent define untouchable numbers, relay their sequence, known results and open conjectures from number theory references, and distinguish them from related concepts such as aliquot sequences and perfect numbers.
Bundle → Layer → Finding → Questions Filled
4 bundles · 8 layers · 8 findings · 16 questions
Define Definition.
Definition
Definition.
Definition
Definition.
- What is an untouchable number, and how is it checked? definition
- Is the question about untouchable numbers or another divisor-based class? boundary
Sequence
Sequence.
Sequence
Sequence.
- What are the first untouchable numbers, and where is the sequence catalogued? measurement
- Which entry fits aliquot sum? action
Results Results and conjectures.
Theorems
Known results.
Theorems
Theorems.
- What did Erdos prove about infinitude, and what density results exist? provenance
- Which references are standard? provenance
Conjectures
Conjectures.
Conjectures
Conjectures.
- Why does the uniqueness of 5 follow from a strong Goldbach conjecture? provenance
- Which entry fits Goldbach conjecture? action
Compute Computation.
Algorithms
Algorithms.
Algorithms
Algorithms.
- How are untouchable numbers computed, and how far have they been enumerated? provenance
- Which sources are cited? provenance
Related
Related classes.
Related
Related.
- How do untouchable numbers relate to aliquot sequences, perfect and amicable numbers? provenance
- Which entry fits aliquot sequence? action
Context History.
History
History.
History
History.
- When were untouchable numbers introduced and studied? provenance
- Which entry fits the history of number theory? action
Recreational
Recreational mathematics.
Recreational
Recreational.
- How do untouchable numbers appear in recreational mathematics? provenance
- Which entry fits recreational mathematics? 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 cannot be expressed as the sum of all the proper divisors of any positive integer, that is, one that is not the aliquot sum of any number, the sequence beginning 2, 5, 52, 88, 96, 120, 124, 146; Erdos proved there are infinitely many untouchable numbers, 5 is conjectured to be the only odd one, a claim that follows from a strong form of the Goldbach conjecture, and the numbers are catalogued in the OEIS.
Why it exists Filled
Let an agent define untouchable numbers, relay their sequence, known results and open conjectures from number theory references, and distinguish them from related concepts such as aliquot sequences and perfect numbers.
Distinguishing features Filled
- Defined via aliquot sums
- Infinitely many
- Link to Goldbach conjecture
- Mostly even
What robots and AI may and may not do Filled
Must not
- State a number is untouchable without a proof or a cited verified result.
- Present the infinitude result without naming the proof it rests on.
- Confuse untouchable numbers with other aliquot sequence notions such as amicable or perfect numbers.
- Publish computed sequence terms as verified when the search bound is unstated.
Only with a human decision
- Publishing a claimed new result about the sequence.
May
- State the defining property of untouchable numbers and the start of the sequence.
- Test whether a given number is untouchable within stated computational limits.
Moral aspects Filled
- Published number sequences are reused by others, so unverified terms spread errors.
- Credit for mathematical results belongs to those who proved them.
Who is affected
- Mathematicians
- Students
- Maintainers of sequence databases
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
- aliquot sum - the sum of proper divisors
- Goldbach conjecture - through the odd case
- OEIS - as A005114
What else AI and robots need to interact with it Filled
Identity and identifiers required Filled
- Vercy registry: vr.tr.untouchable-number
- Wikidata: Q1193504 (https://www.wikidata.org/wiki/Q1193504)
- OEIS: A005114
Direct properties not applicable Not applicable
- smallest: 2 number
- only known odd: 5 number - conjectured unique
Plane XCT: no invented physical properties.
Recognition optional Filled
- Integers that are no number aliquot sum
- Sequence OEIS A005114
- Perfect numbers equal their own aliquot sum; abundant numbers are less than it
- The sequence 2, 5, 52, 88, 96, 120, 124, 146, ...
Capabilities and actions required Filled
- define and list
- relay known results
- relay conjectures
- distinguish related concepts
Hazards and failure modes required Filled
- Confusing with other divisor-based classes
- Stating conjectures as theorems
- Computation errors
Standards and interfaces required Filled
- No regulation; mathematical conventions
Context of use required Filled
- Not applicable; a class of integers.
- untouchable numbers in general
- the odd untouchable number 5
- related classes: aliquot sequences, perfect, abundant and deficient numbers
Sources Filled
- Wikidata item Q1193504: untouchable number - identity and sense of the item
- Wikipedia: Untouchable number - general description of the item
Open questions
- Is 5 the only odd untouchable number?
- How should OEIS and literature be linked?
- What is the asymptotic density of untouchable numbers?
Machine files
Provenance
thing registry research (pass 2) · unreviewed
Built from: models/things/publications/thing-q1193504/spec.json