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.
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 NERFINISHED ⓘ |
| 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
NERFINISHED
ⓘ
Egyptian fractions NERFINISHED ⓘ Erdős–Straus conjecture NERFINISHED ⓘ Euclid–Mullin sequence ⓘ unit fractions ⓘ |
| usedFor | Egyptian fraction representations of 1 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.