Sylvester sequence
E571006
The Sylvester sequence is an integer sequence defined recursively where each term is one more than the product of all previous terms, yielding rapidly growing, pairwise coprime numbers closely related to Egyptian fraction representations.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Sylvester sequence canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6149924 — 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: Sylvester sequence Context triple: [James Joseph Sylvester, notableWork, Sylvester sequence]
-
A.
Ulam sequence
The Ulam sequence is an integer sequence starting with 1 and 2 in which each subsequent term is the smallest integer that can be written uniquely as the sum of two distinct earlier terms.
-
B.
Look-and-say sequence
The look-and-say sequence is a famous integer sequence where each term is generated by verbally describing the digits of the previous term, studied for its surprising combinatorial and growth properties.
-
C.
Pisano period
The Pisano period is the repeating cycle length of Fibonacci numbers when taken modulo a given integer.
-
D.
Lucas sequences
Lucas sequences are a family of integer sequences defined by the same type of second-order linear recurrence as the Fibonacci numbers but with more general initial conditions, encompassing the Fibonacci sequence as a special case.
-
E.
Fibonacci sequence
The Fibonacci sequence is an infinite series of numbers where each term is the sum of the two preceding ones, widely used in mathematics, art, and design due to its connection with the golden ratio and natural growth patterns.
- 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: Sylvester sequence Target entity description: The Sylvester sequence is an integer sequence defined recursively where each term is one more than the product of all previous terms, yielding rapidly growing, pairwise coprime numbers closely related to Egyptian fraction representations.
-
A.
Ulam sequence
The Ulam sequence is an integer sequence starting with 1 and 2 in which each subsequent term is the smallest integer that can be written uniquely as the sum of two distinct earlier terms.
-
B.
Look-and-say sequence
The look-and-say sequence is a famous integer sequence where each term is generated by verbally describing the digits of the previous term, studied for its surprising combinatorial and growth properties.
-
C.
Pisano period
The Pisano period is the repeating cycle length of Fibonacci numbers when taken modulo a given integer.
-
D.
Lucas sequences
Lucas sequences are a family of integer sequences defined by the same type of second-order linear recurrence as the Fibonacci numbers but with more general initial conditions, encompassing the Fibonacci sequence as a special case.
-
E.
Fibonacci sequence
The Fibonacci sequence is an infinite series of numbers where each term is the sum of the two preceding ones, widely used in mathematics, art, and design due to its connection with the golden ratio and natural growth patterns.
- F. None of above. chosen
Statements (42)
| Predicate | Object |
|---|---|
| instanceOf |
integer sequence
ⓘ
mathematical sequence ⓘ number theory concept ⓘ |
| alternativeRecurrence | a_{n+1} = a_n^2 - a_n + 1 for n ≥ 1 ⓘ |
| appearsIn |
research on Egyptian fraction decompositions of 1
ⓘ
studies of rapidly growing integer sequences ⓘ |
| definedByRecurrence |
a_1 = 2
ⓘ
a_{n+1} = 1 + a_1 a_2 \cdots a_n ⓘ |
| field |
combinatorics
ⓘ
number theory ⓘ |
| formula | \sum_{n=1}^{\infty} 1/a_n = 1 ⓘ |
| growthRate | superexponential ⓘ |
| hasClosedFormLikeRelation | a_{n+1} - 1 = \prod_{k=1}^{n} a_k ⓘ |
| hasFifthTerm | 1807 ⓘ |
| hasFirstTerm | 2 ⓘ |
| hasFourthTerm | 43 ⓘ |
| hasKeyword |
pairwise coprime
ⓘ
recursive definition ⓘ unit fraction decomposition ⓘ |
| hasOEISId | A000058 ⓘ |
| hasSecondTerm | 3 ⓘ |
| hasSixthTerm | 3263443 ⓘ |
| hasThirdTerm | 7 ⓘ |
| namedAfter | James Joseph Sylvester ⓘ |
| namedEntityType | mathematical object ⓘ |
| property |
all terms are integers greater than 1
ⓘ
each term is greater than the square of the previous term minus the previous term ⓘ each term is one more than the product of all previous terms ⓘ no term divides another term ⓘ partial sums of reciprocals are strictly increasing and less than 1 ⓘ product of first n terms equals a_{n+1} - 1 ⓘ sequence grows faster than any geometric progression ⓘ sequence is infinite ⓘ sum of reciprocals of all terms equals 1 ⓘ terms are pairwise coprime ⓘ terms are strictly increasing ⓘ |
| relatedTo |
Egyptian fraction greedy algorithm
ⓘ
Egyptian fractions ⓘ
linked to:
Egyptian mathematics
Erdős–Straus conjecture ⓘ Euclid–Mullin sequence ⓘ unit fractions ⓘ |
| usedFor | Egyptian fraction representations of 1 ⓘ |
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: Sylvester sequence Description of subject: The Sylvester sequence is an integer sequence defined recursively where each term is one more than the product of all previous terms, yielding rapidly growing, pairwise coprime numbers closely related to Egyptian fraction representations.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.