A002210 Decimal expansion of Khinchin's constant.

2, 6, 8, 5, 4, 5, 2, 0, 0, 1, 0, 6, 5, 3, 0, 6, 4, 4, 5, 3, 0, 9, 7, 1, 4, 8, 3, 5, 4, 8, 1, 7, 9, 5, 6, 9, 3, 8, 2, 0, 3, 8, 2, 2, 9, 3, 9, 9, 4, 4, 6, 2, 9, 5, 3, 0, 5, 1, 1, 5, 2, 3, 4, 5, 5, 5, 7, 2, 1, 8, 8, 5, 9, 5, 3, 7, 1, 5, 2, 0, 0, 2, 8, 0, 1, 1, 4, 1, 1, 7, 4, 9, 3, 1, 8, 4, 7, 6, 9, 7, 9, 9, 5, 1, 5

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• cons
• nice

Comments

A. Ya. Khinchin is the preferred spelling.

Carles Simó, Oct 11 2016, reports that he has computed 10^6 terms of this sequence (see links). - N. J. A. Sloane, Nov 04 2016

Named after the Soviet mathematician Aleksandr Yakovlevich Khinchin (1894 - 1959). - Amiram Eldar, Aug 19 2020

Examples

			2.685452001065306445309714835481795693820382293994462953051152345557218...
		

References

• S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, pp. 59-65.
• A. Ya. Khintchin, Continued Fractions, Groningen: Noordhoff, 1963.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• I. Vardi, Computational Recreations in Mathematica. Addison-Wesley, Redwood City, CA, 1991, p. 164.

Links

• Harry J. Smith, Table of n, a(n) for n = 1..1200
• D. H. Bailey, J. M. Borwein & R. E. Crandall, On the Khintchine Constant
• Ph. Flajolet and I. Vardi, Zeta function expansions of some classical constants
• E. Fontich, C. Simó, A. Vieiro, On the "hidden" harmonics associated to best approximants due to quasiperiodicity in splitting phenomena, Regular and Chaotic Dynamics (2018), Pleiades Publishing, Vol. 23, Issue 6, 638-653.
• Brady Haran and Tony Padilla, Six Sequences, Numberphile video, 2013.
• A. Khintchine, Metrische kettenbruchprobleme, Compositio Mathematica, Vol. 1 (1935), pp. 361-382.
• A. Khintchine, Zur metrischen Kettenbruchtheorie, Compositio Mathematica, Vol. 3 (1936), pp. 276-285.
• Christian Perfect, Integer sequence reviews on Numberphile (or vice versa), 2013.
• Simon Plouffe, 110000 digits of the Khintchine constant
• Simon Plouffe, Khinchin constant to 1024 digits
• D. Shanks and J. W. Wrench, Jr., Khintchine's constant, Amer. Math. Monthly, 66 (1959), 276-279.
• Carles Simó, Computation of 10^6 digits of Khintchine's constant
• Carles Simó, Computation of 10^6 digits of Khintchine's constant [Cached copy, with permission]
• Carles Simó, 10^6 digits of Khintchine's constant [Cached copy, with permission]
• Eric Weisstein's World of Mathematics, Continued Fraction
• Eric Weisstein's World of Mathematics, Khinchin's Constant
• Eric Weisstein's World of Mathematics, Khinchin's Constant Digits
• Thomas Wieting, A Khinchin Sequence, Proc. Amer. Math. Soc. 136 (2008), 815-824.
• Wikipedia, Khinchin's constant
• J. W. Wrench, Further evaluation of Khintchine's constant, Math. Comp., 14 (1960), 370-371.

Crossrefs

Cf. A002211, A247038.

Programs

Mathematica

RealDigits[N[Khinchin, 100]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jun 18 2009 *)

Python

from mpmath import mp, khinchin
mp.dps = 106
print([int(k) for k in list(str(khinchin).replace('.', ''))[:-1]]) # Indranil Ghosh, Jul 08 2017

Formula

From Amiram Eldar, Aug 19 2020: (Start)

Equal Product_{k>=1} (1 + 1/(k*(k+2)))^log_2(k).

Equals exp(A247038/log(2)). (End)

Extensions

Pari code removed by D. S. McNeil, Dec 26 2010

Spelling of Kninchin's name normalized by N. J. A. Sloane, Jul 12 2024