abundant number
Let an agent handle abundant numbers by definition, examples, related classes and open problems.
Research draft, second pass
A second pass drafted this model: the structure a model of this thing needs, and what is known about it in the world. The line under this one says how the second half was obtained - researched against sources, or recalled without web access, in which case nothing here was read anywhere and every claim is a lead to verify. Unreviewed either way.
written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify
Researched by: Claude
Purpose and description
Let an agent handle abundant numbers by definition, examples, related classes and open problems.
A positive integer for which the sum of its proper divisors exceeds the number itself, such as 12 (1 + 2 + 3 + 4 + 6 = 16); related classes include primitive abundant, weird and quasiperfect numbers.
What it is for: Number theory and classification of integers.
It can be test whether a number is abundant; list abundant numbers; relate them to perfect and deficient numbers; describe related open problems.
Distinguishing features
Divisor sum exceeds the number
Infinitely many
Most are even
Related to weird numbers
What it looks like
Integers such as 12, 18, 20, 24 and 30.
How it is recognised
Sum of proper divisors greater than the number
Smallest odd abundant number is 945
Perfect numbers have equal sums
Related models
is a kind of - category
is contrasted with - classes
is defined by - concept
is studied in - field
In practice
Families and kinds
primitive abundant numbers
odd abundant numbers
weird numbers
quasiperfect numbers (hypothetical)
Identifiers
OEIS A005101 integer sequence
Failure modes and hazards
Counting the number itself among proper divisors
Stating open problems as solved
Also called
Where this came from
wikidata · CC0 1.0
Drafted structure
Bundle to layer to finding to question, as the second pass will find it: 4 bundles · 8 layers · 8 findings · 16 questions.
Test Is it abundant.
Divisor sums decide.
Check
Sum of divisors.
Check
Abundance test.
- Is this number abundant, perfect or deficient? boundary
- What is its abundance? measurement
Primitive
No abundant divisors.
Primitive
Primitive abundance.
- Is it primitive abundant? boundary
- Which divisors are abundant? definition
Examples Known values.
Examples illustrate.
List
Sequence.
List
List of abundant numbers.
- What are the first abundant numbers? provenance
- Which is the smallest odd one? definition
Weird
Weird numbers.
Weird
Weird numbers.
- What makes a number weird? definition
- Is 70 the smallest? provenance
Theory Results.
Theory gives structure.
Density
How common.
Density
Density.
- What proportion of integers are abundant? measurement
- Which source gives the estimate? provenance
Open
Unsolved.
Open
Open problems.
- Are odd weird numbers or quasiperfect numbers known to exist? provenance
- Is the status stated accurately? boundary
Learning Teaching.
Activities engage learners.
Activities
Exercises.
Activities
Activities.
- Which exercises help learners classify numbers by divisor sums? action
- How can divisors be found quickly? action
History
Origins.
History
History.
- How were abundant numbers classified historically? provenance
- Which mathematicians studied them? provenance
What the second pass must settle
- Should weird numbers be a separate entry?
- How should OEIS data be linked?
- How should quasiperfect numbers be described?