The nth radical root of the denominator of the nth convergent of a number x tends to a constant
(Sloane's A086702) for all but a set of x of measure zero (Lévy 1936, Lehmer 1939). Plouffe called the exponent the Khinchin-Lévy constant. Taking its multiplicative inverse gives (Sloane's A089729).
The plot above shows for the first 500
terms in the continued fractions of
Gauss-Kuzmin-Wirsing Constant, Khinchin's Constant
Finch, S. R. Mathematical Constants. Cambridge, England: Cambridge University Press, p. 156, 2003.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 51, 1983.
Lehmer, D. H. "Note on an Absolute Constant of Khintchine." Amer. Math. Monthly 46, 148-152, 1939.
Lévy, P. "Sur le développement en fraction continue d'un nombre choisi au hasard." Compositio Math. 3, 286-303, 1936. Reprinted in uvres de Paul Lévy, Vol. 6. Paris: Gauthier-Villars, pp. 285-302, 1980.
Rockett, A. M. and Szüsz, P. "The Khintchine-Lévy Theorem for
Sloane, N. J. A. Sequences A086702 and A089729 in "The On-Line Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/.
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