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A002210
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Decimal expansion of Khintchine's constant.
(Formerly M1564 N0609)
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48
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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
(list;
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OFFSET
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1,1
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COMMENTS
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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 Khintchine (1894 - 1959). - Amiram Eldar, Aug 19 2020
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REFERENCES
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S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, pp. 59-65.
A. Ya. Khintchine, 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.
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LINKS
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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.
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FORMULA
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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)
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EXAMPLE
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2.685452001065306445309714835481795693820382293994462953051152345557218...
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MATHEMATICA
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RealDigits[N[Khinchin, 100]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jun 18 2009 *)
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PROG
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(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
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CROSSREFS
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Cf. A002211, A247038.
Sequence in context: A121862 A095677 A011045 * A145500 A247572 A065129
Adjacent sequences: A002207 A002208 A002209 * A002211 A002212 A002213
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KEYWORD
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nonn,cons,nice
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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Pari code removed by D. S. McNeil, Dec 26 2010
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STATUS
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approved
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