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%I M1564 N0609 #93 Oct 16 2022 04:14:46
%S 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,
%T 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,
%U 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
%N Decimal expansion of Khintchine's constant.
%C _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
%C Named after the Soviet mathematician Aleksandr Yakovlevich Khintchine (1894 - 1959). - _Amiram Eldar_, Aug 19 2020
%D S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, pp. 59-65.
%D A. Ya. Khintchine, Continued Fractions, Groningen: Noordhoff, 1963.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D I. Vardi, Computational Recreations in Mathematica. Addison-Wesley, Redwood City, CA, 1991, p. 164.
%H Harry J. Smith, <a href="/A002210/b002210.txt">Table of n, a(n) for n = 1..1200</a>
%H D. H. Bailey, J. M. Borwein & R. E. Crandall, <a href="https://www.ams.org/mcom/1997-66-217/S0025-5718-97-00800-4/S0025-5718-97-00800-4.pdf">On the Khintchine Constant</a>
%H Ph. Flajolet and I. Vardi, <a href="http://algo.inria.fr/flajolet/Publications/publist.html">Zeta function expansions of some classical constants</a>
%H E. Fontich, C. Simó, A. Vieiro, <a href="https://doi.org/10.1134/S1560354718060011">On the "hidden" harmonics associated to best approximants due to quasiperiodicity in splitting phenomena</a>, Regular and Chaotic Dynamics (2018), Pleiades Publishing, Vol. 23, Issue 6, 638-653.
%H Brady Haran and Tony Padilla, <a href="http://www.youtube.com/watch?v=VDD6FDhKCYA">Six Sequences</a>, Numberphile video, 2013.
%H A. Khintchine, <a href="http://www.numdam.org/item/?id=CM_1935__1__361_0">Metrische kettenbruchprobleme</a>, Compositio Mathematica, Vol. 1 (1935), pp. 361-382.
%H A. Khintchine, <a href="http://www.numdam.org/item/CM_1936__3__276_0/">Zur metrischen Kettenbruchtheorie</a>, Compositio Mathematica, Vol. 3 (1936), pp. 276-285.
%H Christian Perfect, <a href="http://aperiodical.com/2013/07/integer-sequence-reviews-on-numberphile-or-vice-versa/">Integer sequence reviews on Numberphile (or vice versa)</a>, 2013.
%H Simon Plouffe, <a href="http://www.plouffe.fr/simon/constants/khintchine.txt">110000 digits of the Khintchine constant</a>
%H Simon Plouffe, <a href="http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap50.html">Khinchin constant to 1024 digits</a>
%H D. Shanks and J. W. Wrench, Jr., <a href="http://www.jstor.org/stable/2309633">Khintchine's constant</a>, Amer. Math. Monthly, 66 (1959), 276-279.
%H Carles Simó, <a href="http://www.maia.ub.es/dsg/khinchin">Computation of 10^6 digits of Khintchine's constant</a>
%H Carles Simó, <a href="/A002210/a002210.png">Computation of 10^6 digits of Khintchine's constant</a> [Cached copy, with permission]
%H Carles Simó, <a href="/A002210/a002210.txt">10^6 digits of Khintchine's constant</a> [Cached copy, with permission]
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ContinuedFraction.html">Continued Fraction</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/KhinchinsConstant.html">Khinchin's Constant</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/KhinchinsConstantDigits.html">Khinchin's Constant Digits</a>
%H Thomas Wieting, <a href="http://dx.doi.org/10.1090/S0002-9939-07-09202-7">A Khinchin Sequence</a>, Proc. Amer. Math. Soc. 136 (2008), 815-824.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Khinchin%27s_constant">Khinchin's constant</a>
%H J. W. Wrench, <a href="http://dx.doi.org/10.1090/S0025-5718-1960-0170455-1">Further evaluation of Khintchine's constant</a>, Math. Comp., 14 (1960), 370-371.
%F From _Amiram Eldar_, Aug 19 2020: (Start)
%F Equal Product_{k>=1} (1 + 1/(k*(k+2)))^log_2(k).
%F Equals exp(A247038/log(2)). (End)
%e 2.685452001065306445309714835481795693820382293994462953051152345557218...
%t RealDigits[N[Khinchin, 100]][[1]] (* _Vladimir Joseph Stephan Orlovsky_, Jun 18 2009 *)
%o (Python)
%o from mpmath import mp, khinchin
%o mp.dps = 106
%o print([int(k) for k in list(str(khinchin).replace('.', ''))[:-1]]) # _Indranil Ghosh_, Jul 08 2017
%Y Cf. A002211, A247038.
%K nonn,cons,nice
%O 1,1
%A _N. J. A. Sloane_
%E Pari code removed by _D. S. McNeil_, Dec 26 2010
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