highly composite numbers
E355438
Highly composite numbers are positive integers that have more divisors than any smaller positive integer, extensively studied and characterized by Srinivasa Ramanujan.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Highly Composite Numbers | 1 |
| highly composite numbers canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T3410522 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: highly composite numbers Context triple: [Srinivasa Ramanujan, notableWork, highly composite numbers]
-
A.
Hardy–Ramanujan asymptotic formula
The Hardy–Ramanujan asymptotic formula is a landmark result in number theory that gives an approximate expression for the partition function p(n), describing how the number of integer partitions of n grows rapidly with n.
-
B.
Chebyshev functions
Chebyshev functions are arithmetic functions in number theory that encode information about the distribution of prime numbers and play a key role in analytic approaches to the prime number theorem.
-
C.
Hardy–Littlewood circle method
The Hardy–Littlewood circle method is a powerful analytic number theory technique that uses complex analysis and Fourier series to study additive problems such as Waring’s problem and the Goldbach conjecture.
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D.
Hardy–Littlewood conjectures
The Hardy–Littlewood conjectures are a collection of influential unproven hypotheses in analytic number theory that generalize the prime number theorem to describe the distribution of prime numbers and prime constellations.
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E.
Jordan’s totient functions
Jordan’s totient functions are a family of arithmetic functions in number theory that generalize Euler’s totient function to count k-tuples of integers modulo n with certain coprimality conditions.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: highly composite numbers Target entity description: Highly composite numbers are positive integers that have more divisors than any smaller positive integer, extensively studied and characterized by Srinivasa Ramanujan.
-
A.
Hardy–Ramanujan asymptotic formula
The Hardy–Ramanujan asymptotic formula is a landmark result in number theory that gives an approximate expression for the partition function p(n), describing how the number of integer partitions of n grows rapidly with n.
-
B.
Chebyshev functions
Chebyshev functions are arithmetic functions in number theory that encode information about the distribution of prime numbers and play a key role in analytic approaches to the prime number theorem.
-
C.
Hardy–Littlewood circle method
The Hardy–Littlewood circle method is a powerful analytic number theory technique that uses complex analysis and Fourier series to study additive problems such as Waring’s problem and the Goldbach conjecture.
-
D.
Hardy–Littlewood conjectures
The Hardy–Littlewood conjectures are a collection of influential unproven hypotheses in analytic number theory that generalize the prime number theorem to describe the distribution of prime numbers and prime constellations.
-
E.
Jordan’s totient functions
Jordan’s totient functions are a family of arithmetic functions in number theory that generalize Euler’s totient function to count k-tuples of integers modulo n with certain coprimality conditions.
- F. None of above. chosen
Statements (54)
| Predicate | Object |
|---|---|
| instanceOf |
integer sequence
ⓘ
mathematical concept ⓘ |
| alternativeName | anti-prime ⓘ |
| application |
used in design of measurement systems with many divisors
ⓘ
used in problems involving highly divisible numbers ⓘ used in scheduling and partitioning problems ⓘ |
| classification | subset of positive integers ⓘ |
| definition | positive integer that has more divisors than any smaller positive integer ⓘ |
| eighthTerm | 48 ⓘ |
| field | number theory ⓘ |
| fifthTerm | 12 ⓘ |
| firstTerm | 1 ⓘ |
| fourthTerm | 6 ⓘ |
| growthProperty | terms grow faster than linearly ⓘ |
| hasDivisorsCount |
1 has 1 divisor
ⓘ
12 has 6 divisors ⓘ 120 has 16 divisors ⓘ 2 has 2 divisors ⓘ 24 has 8 divisors ⓘ 36 has 9 divisors ⓘ 4 has 3 divisors ⓘ 48 has 10 divisors ⓘ 6 has 4 divisors ⓘ 60 has 12 divisors ⓘ |
| hasOpenProblems |
asymptotic behavior of counting function
ⓘ
distribution of highly composite numbers ⓘ |
| introducedBy | Srinivasa Ramanujan ⓘ |
| ninthTerm | 60 ⓘ |
| OEISSequence | A002182 ⓘ |
| property |
all highly composite numbers are composite except 1
ⓘ
defined using the divisor function d(n) ⓘ each term has a record number of divisors ⓘ prime exponents in factorization form a non-increasing sequence ⓘ prime factorization uses small primes with non-increasing exponents ⓘ related to divisor-maximizing problems ⓘ sequence is strictly increasing ⓘ tend to be very dense in divisors ⓘ |
| publication |
Highly Composite Numbers
ⓘ
linked to:
highly composite numbers
|
| publicationAuthor | Srinivasa Ramanujan ⓘ |
| publicationYear | 1915 ⓘ |
| relatedTo |
Ramanujan
ⓘ
linked to:
Srinivasa Ramanujan
abundant numbers ⓘ colossally abundant numbers ⓘ divisor function ⓘ highly composite k-tuples ⓘ superabundant numbers ⓘ |
| secondTerm | 2 ⓘ |
| seventhTerm | 36 ⓘ |
| sixthTerm | 24 ⓘ |
| studiedBy | Srinivasa Ramanujan ⓘ |
| symbolicDescription | n is highly composite if d(n) > d(k) for all positive integers k < n ⓘ |
| tenthTerm | 120 ⓘ |
| thirdTerm | 4 ⓘ |
| yearCharacterized | 1915 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: highly composite numbers Description of subject: Highly composite numbers are positive integers that have more divisors than any smaller positive integer, extensively studied and characterized by Srinivasa Ramanujan.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject linked to:
highly composite numbers
linked to: highly composite numbers