deficient number
Let an agent explain deficient numbers and their relation to perfect and abundant numbers, test whether a number is deficient, describe known results and distribution, and support number theory learning.
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 explain deficient numbers and their relation to perfect and abundant numbers, test whether a number is deficient, describe known results and distribution, and support number theory learning.
A positive integer for which the sum of its proper divisors is less than the number itself, such as 8, whose proper divisors 1, 2 and 4 sum to 7, in contrast to perfect numbers where the sum equals the number and abundant numbers where it exceeds it; all prime numbers and prime powers are deficient, deficient numbers form the majority of integers, and the classification dates to ancient Greek number theory.
What it is for: Integers with small divisor sums.
It can be explain the definition; test numbers; describe results and distribution; support learning.
Distinguishing features
Aliquot sum classification
Includes primes
Majority of integers
Ancient origin
What it looks like
Not physical; a class of integers.
How it is recognised
Sum of proper divisors less than the number
Includes all primes
Perfect numbers equal their divisor sum; abundant exceed it
Related models
is a kind of - category
is related to - another aliquot classification
is related to - another integer sequence entry
is related to - numbers that are prime or composite
In practice
Families and kinds
prime numbers as deficient numbers
prime powers
deficient composite numbers
almost perfect numbers
deficient numbers in aliquot sequences
Standards and regulation
Mathematical definitions and notation
Sequence database conventions
Failure modes and hazards
Including the number itself in the divisor sum
Confusing deficient with abundant
Overstating what is proven
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.
Understand What deficient numbers are.
Mathematics.
Definition
Definition and examples.
Definition
Definition.
- What makes a number deficient, and how do deficient, perfect and abundant numbers partition the integers? definition
- Which entry fits perfect or abundant numbers? action
Properties
Properties.
Properties
Properties.
- Why are all primes and prime powers deficient, and what are almost perfect numbers? definition
- Which results about divisors of deficient numbers are known? provenance
Compute Testing numbers.
Practice.
Test
Testing deficiency.
Test
Testing.
- Is this number deficient, and what is its aliquot sum? action
- Which entry fits divisor functions? action
Distribution
Distribution.
Distribution
Distribution.
- What proportion of integers are deficient, and how is the density known? provenance
- Which references are standard? provenance
Connect Connections.
Theory.
Aliquot
Aliquot sequences.
Aliquot
Aliquot.
- How do deficient numbers behave in aliquot sequences? definition
- Which entry fits aliquot sequences? action
Open
Open problems.
Open
Open problems.
- Which open problems involve deficient numbers and the divisor sum function? boundary
- Which findings are recent? provenance
Learn History and teaching.
Education.
History
History.
History
History.
- How did Nicomachus and other ancient mathematicians classify deficient, perfect and abundant numbers? provenance
- Which entry fits history of number theory? action
Teach
Teaching.
Teach
Teaching.
- How can divisor classifications be used to teach factors? action
- Which misconceptions arise? provenance
What the second pass must settle
- Should abundant and perfect numbers be linked as separate entries?
- How should sequence databases be linked?
- How should open problems be documented?