Concept
In number theory, Sylvester's sequence is an integer sequence in which each term is the product of the previous terms, plus one. The first few terms of the sequence are 2, 3, 7, 43, 1807, 3263443, 10650056950807, 113423713055421844361000443 . 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. 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 terms. 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. Formally, Sylvester's sequence can be defined by the formula The product of the empty set is 1, so s0 = 2. Alternatively, one may define the sequence by the recurrence with s0 = 2. It is straightforward to show by induction that this is equivalent to the other definition. The Sylvester numbers grow doubly exponentially as a function of n. Specifically, it can be shown that for a number E that is approximately 1.26408473530530... . This formula has the effect of the following algorithm: s0 is the nearest integer to E 2; s1 is the nearest integer to E 4; s2 is the nearest integer to E 8; for sn, take E 2, square it n more times, and take the nearest integer. This would only be a practical algorithm if we had a better way of calculating E to the requisite number of places than calculating sn and taking its repeated square root.
