Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In number theory, Sylvester's sequence is a sequence of integers in which each member of the sequence is the product of the previous members, plus one. The first few terms of the sequence are: 2, 3, 7, 43, 1807, 3263443, 10650056950807, 113423713055421844361000443 (sequence A000058 in OEIS). Sylvester's sequence is named after James Joseph Sylvester, who first investigated it in 1880. 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 with the same sum. 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, but, due to the rapid growth of the sequence, complete prime factorizations are known only for a few of its members. Values derived from this sequence have also been used to construct finite Egyptian fraction representations of 1, Sasakian Einstein manifolds, and hard instances for online algorithms.