Sylvester's sequence
Doubly exponential integer sequence / From Wikipedia, the free encyclopedia
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In number theory, Sylvester's sequence is an integer sequence in which each term is the product of the previous terms, plus one. Its first few terms are
- 2, 3, 7, 43, 1807, 3263443, 10650056950807, 113423713055421844361000443 (sequence A000058 in the OEIS).
Sylvester's sequence is named after James Joseph Sylvester, who first investigated it in 1880.[1] Its values grow doubly exponentially, and the sum of its reciprocals forms a series of unit fractions that converges to 1 more rapidly than any other series of unit fractions.[2] The recurrence by which it is defined allows the numbers in the sequence to be factored more easily than other numbers of the same magnitude,[3] but, due to the rapid growth of the sequence, complete prime factorizations are known only for a few of its terms.[4] Values derived from this sequence have also been used to construct finite Egyptian fraction representations of 1, Sasakian Einstein manifolds,[5] and hard instances for online algorithms.[6]