Sylvester's sequence
[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.
Alternatively, one may define the sequence by the recurrence[3] It is straightforward to show by induction that this is equivalent to the other definition.
, but they can also be defined by a product formula very similar to that defining Sylvester's sequence:[12] The unit fractions formed by the reciprocals of the values in Sylvester's sequence generate an infinite series:[13] The partial sums of this series have a simple form, which is already in lowest terms.
[14] This may be proved by induction, or more directly by noting that the recursion implies that so the sum telescopes[14] Since this sequence of partial sums (sj − 2)/(sj − 1) converges to one, the overall series forms an infinite Egyptian fraction representation of the number one: One can find finite Egyptian fraction representations of one, of any length, by truncating this series and subtracting one from the last denominator: The sum of the first k terms of the infinite series provides the closest possible underestimate of 1 by any k-term Egyptian fraction.
It is possible to interpret the Sylvester sequence as the result of a greedy algorithm for Egyptian fractions, that at each step chooses the smallest possible denominator that makes the partial sum of the series be less than one.
[17] Erdős & Graham (1980) conjectured that, in results of this type, the inequality bounding the growth of the sequence could be replaced by a weaker condition,[18] Badea (1995) surveys progress related to this conjecture; see also Brown (1979).
They show that the number of distinct Sasakian Einstein metrics on a topological sphere of dimension 2n − 1 is at least proportional to sn and hence has double exponential growth with n.[5] As Galambos & Woeginger (1995) describe, Brown (1979) and Liang (1980) used values derived from Sylvester's sequence to construct lower bound examples for online bin packing algorithms.
[6] Seiden & Woeginger (2005) similarly use the sequence to lower bound the performance of a two-dimensional cutting stock algorithm.
Solutions to Znám's problem have applications to the classification of surface singularities (Brenton and Hill 1988) and to the theory of nondeterministic finite automata.
[26] Curtiss (1922) describes an application of the closest approximations to one by k-term sums of unit fractions, in lower-bounding the number of divisors of any perfect number, and Miller (1919) uses the same property to upper bound the size of certain groups.
