Sylvester's sequence

In number theory, Sylvester's sequence is an integer sequence in which each member of the sequence is the product of the previous members, plus one. The first few terms of the sequence are:

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 number of terms. 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.

Formally, Sylvester's sequence can be defined by the formula

The product of an empty set is 1, so s₀ = 2.

Alternatively, one may define the sequence by the recurrence

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 1.264084735305302. This formula has the effect of the following algorithm:

This would only be a practical algorithm if we had a better way of calculating E to the requisite number of places than calculating s_n and taking its repeated square root.

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