Sylvester's sequence

Graphical demonstration of the convergence of the sum 1/2 + 1/3 + 1/7 + 1/43 + ... to 1. Each row of k squares of side length 1/k has total area 1/k, and all the squares together exactly cover a larger square with area 1. Squares with side lengths 1/1807 or smaller are too small to see in the figure and are not shown.

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]

  1. ^ Sylvester (1880).
  2. ^ Cite error: The named reference closest was invoked but never defined (see the help page).
  3. ^ Vardi (1991).
  4. ^ Cite error: The named reference factorizations was invoked but never defined (see the help page).
  5. ^ Boyer, Galicki & Kollár (2005).
  6. ^ Galambos & Woeginger (1995); Brown (1979); Liang (1980).