Sylvester's Sequence - Uniqueness of Quickly Growing Series With Rational Sums

Uniqueness of Quickly Growing Series With Rational Sums

As Sylvester himself observed, Sylvester's sequence seems to be unique in having such quickly growing values, while simultaneously having a series of reciprocals that converges to a rational number.

To make this more precise, it follows from results of Badea (1993) that, if a sequence of integers grows quickly enough that

and if the series

converges to a rational number A, then, for all n after some point, this sequence must be defined by the same recurrence

that can be used to define Sylvester's sequence.

Erdős (1980) conjectured that, in results of this type, the inequality bounding the growth of the sequence could be replaced by a weaker condition,

Badea (1995) surveys progress related to this conjecture; see also Brown (1979).

Read more about this topic:  Sylvester's Sequence

Famous quotes containing the words uniqueness of, uniqueness, quickly, growing, series, rational and/or sums:

    Somehow we have been taught to believe that the experiences of girls and women are not important in the study and understanding of human behavior. If we know men, then we know all of humankind. These prevalent cultural attitudes totally deny the uniqueness of the female experience, limiting the development of girls and women and depriving a needy world of the gifts, talents, and resources our daughters have to offer.
    Jeanne Elium (20th century)

    Until now when we have started to talk about the uniqueness of America we have almost always ended by comparing ourselves to Europe. Toward her we have felt all the attraction and repulsions of Oedipus.
    Daniel J. Boorstin (b. 1914)

    Out of power, Marxism can develop critical intelligence; in power, it quickly becomes stupid.
    Mason Cooley (b. 1927)

    I am not afraid of the priests in the long-run. Scientific method is the white ant which will slowly but surely destroy their fortifications. And the importance of scientific method in modern practical life—always growing and increasing—is the guarantee for the gradual emancipation of the ignorant upper and lower classes, the former of whom especially are the strength of the priests.
    Thomas Henry Huxley (1825–95)

    If the technology cannot shoulder the entire burden of strategic change, it nevertheless can set into motion a series of dynamics that present an important challenge to imperative control and the industrial division of labor. The more blurred the distinction between what workers know and what managers know, the more fragile and pointless any traditional relationships of domination and subordination between them will become.
    Shoshana Zuboff (b. 1951)

    [I]n Great-Britain it is said that their constitution relies on the house of commons for honesty, and the lords for wisdom; which would be a rational reliance if honesty were to be bought with money, and if wisdom were hereditary.
    Thomas Jefferson (1743–1826)

    If God lived on earth, people would break his windows.
    Jewish proverb, quoted in Claud Cockburn, Cockburn Sums Up, epigraph (1981)