harshad number
Let an agent define harshad numbers, test and generate them in any base, present known results and open questions, and support recreational mathematics 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 define harshad numbers, test and generate them in any base, present known results and open questions, and support recreational mathematics learning.
A positive integer that is divisible by the sum of its digits in a given base, such as 18, whose digits sum to 9, also called a Niven number, with variants such as multiple harshad numbers and base-dependent properties; harshad numbers are studied in recreational and elementary number theory, where questions about consecutive harshad numbers and their distribution have been answered or remain open.
What it is for: A class of integers in number theory.
It can be define and test harshad numbers; generate them in a base; present results and open problems; support learning.
Distinguishing features
Digit-sum divisibility
Base-dependent
Recreational interest
Known results
What it looks like
Not physical; integers with a divisibility property.
How it is recognised
Divisible by its digit sum
Base-dependent
Self-descriptive numbers are a different concept despite the alias
Related models
is a kind of - category
is related to - base dependence
is related to - digit sums
is related to - another recreational number class
In practice
Families and kinds
harshad numbers in base ten
harshad numbers in other bases
multiple harshad numbers
consecutive harshad numbers
all-harshad numbers
Standards and regulation
Mathematical notation conventions
Failure modes and hazards
Confusing with self-descriptive or Niven variants
Forgetting base dependence
Overstating 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.
Define What harshad numbers are.
Definition.
Definition
Definition and examples.
Definition
Definition.
- What is a harshad number, and why are 18 and 21 harshad while 11 is not? definition
- Why does the property depend on the base? definition
Test
Testing a number.
Test
Testing.
- Is this number harshad in the given base? action
- How can harshad numbers be generated by computer? action
Theory Results and problems.
Mathematics.
Results
Known results.
Results
Results.
- What is known about consecutive harshad numbers, multiple harshad numbers and their density? provenance
- What are all-harshad numbers? definition
Open
Open questions.
Open
Open questions.
- Which questions about harshad numbers remain open? provenance
- Which sequence databases record them? provenance
History Origins.
Context.
Naming
Names and history.
Naming
Naming.
- Where do the names harshad and Niven come from, and who studied these numbers? provenance
- Which references are standard? provenance
Related
Related classes.
Related
Related classes.
- How do harshad numbers relate to digit-based classes such as self-descriptive and Smith numbers? definition
- Which entry fits a related class? action
Learn Teaching.
Education.
Teach
Teaching with harshad numbers.
Teach
Teaching.
- How can harshad numbers be used to teach divisibility and programming? action
- Which misconceptions arise? provenance
Puzzles
Puzzles.
Puzzles
Puzzles.
- Which puzzles and exercises involve harshad numbers? action
- How can they be extended? action
What the second pass must settle
- Should recreational number classes be a single entry?
- How should integer sequence databases be linked?
- How should open problems be tracked?