A002211 -id:A002211 - cp's OEIS frontend

A048613 Number of terms (excluding the first) of A002211 for which the geometric mean produces progressively better approximations to Khinchin's constant (itself).

1, 2, 3, 15, 23, 26, 81, 104, 109, 111, 120, 127, 135, 136, 141, 142, 144, 145, 146, 147, 148, 5920, 5943, 8381, 8401, 89953, 91368, 267848, 353014

Offset: 1

Hans Havermann

a(30) > 969679. - Hans Havermann, Jul 14 2024

Hans Havermann, Simple Continued Fraction Expansion of Khinchin's Constant.
Marek Wolf, Computer experiments with Mersenne primes, arXiv preprint arXiv:1112.2412 [math.NT], 2011.

Cf. A002211, A209600.

cf=Rest[ContinuedFraction[Khinchin,200]];r=2;p=N[1,50];Do[p=p*cf[[i]];m=Abs[p^(1/i)-Khinchin];If[mHans Havermann, Jul 16 2024 *)

a(28)-a(29) from Hans Havermann, Jul 14 2024

A002210 Decimal expansion of Khinchin's constant.

Original entry on oeis.org

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

N. J. A. Sloane

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

			2.685452001065306445309714835481795693820382293994462953051152345557218...

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.

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.

Cf. A002211, A247038.

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

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

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)

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

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

A054781 First position of n in continued fraction for Khinchin's constant.

Original entry on oeis.org

2, 1, 10, 47, 4, 34, 76, 65, 119, 11, 104, 27, 103, 110, 675, 80, 1080, 146, 142, 369, 246, 586, 679, 16, 1428, 1621, 1021, 1627, 64, 1342, 799, 157, 409, 506, 1406, 1783, 1445, 206, 3160, 300, 2683, 2037, 4207, 5204, 271, 523, 368, 7892, 2255, 72, 970

Offset: 1

Hans Havermann, May 27 2000

nonn

Indexing of the terms is based on writing a c.f. as [a_1; a_2, a_3, ...]; the more standard convention of [a_0; a_1, a_2, ...] requires subtracting 1 from each term of the sequence.

Smallest positive integers not occurring in the first 106621 terms of the c.f. are 236, 260, 265, 279, 282, 290, 294, 297, 299, ... - Eric W. Weisstein, Oct 01 2011

H. Havermann, Simple Continued Fraction Expansion of Khinchin's Constant
Eric Weisstein's World of Mathematics, Khinchin's Constant Continued Fraction.

Cf. A224851 (= a(n) - 1).

Cf. A002211, A054866, A054870.

a(n) = A224851(n) + 1.

A054866 Incrementally largest terms in the continued fraction for Khinchin's constant.

Original entry on oeis.org

2, 5, 10, 24, 90, 770, 941, 11759, 54097, 231973

Offset: 1

Hans Havermann, May 27 2000

No other high water marks in the first 106621 terms of the c.f. - Eric W. Weisstein, Oct 01 2011

H. Havermann, Simple Continued Fraction Expansion of Khinchin's Constant
Eric Weisstein's World of Mathematics, Khinchin's Constant

Cf. A002211, A054781, A054870.

DeleteDuplicates[ContinuedFraction[Khinchin,10000],GreaterEqual] (* The program generates the first 8 terms of the sequence. *) (* Harvey P. Dale, Sep 11 2024 *)

A054870 Positions of the incrementally largest terms in the continued fraction for Khinchin's constant.

Original entry on oeis.org

1, 4, 11, 16, 24, 105, 1702, 2446, 18996, 60038

Offset: 1

Hans Havermann, May 27 2000

No other high water marks in the first 106621 terms of the c.f. - Eric W. Weisstein, Oct 01 2011

H. Havermann, Simple Continued Fraction Expansion of Khinchin's Constant
Eric Weisstein's World of Mathematics, Khinchin's Constant

Cf. A002211, A054781, A054866.

A127005 Numerators of convergents to Khinchin's constant.

Original entry on oeis.org

2, 3, 8, 43, 51, 94, 239, 333, 572, 2049, 21062, 44173, 65235, 239878, 544991, 13319662, 13864653, 54913621, 123691895, 425989306, 549681201, 975670507, 1525351708, 138257324227, 278040000162, 416297324389, 5273607892830

Offset: 1

Eric W. Weisstein, Jan 02 2007

nonn

			2, 3, 8/3, 43/16, 51/19, 94/35, 239/89, 333/124, 572/213, 2049/763, ...

G. C. Greubel, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Khinchin's Constant
Eric Weisstein's World of Mathematics, Khinchin's Constant Digits
Wikipedia, Khinchin's constant

Cf. A002210, A002211, A127006.

Numerator[Convergents[ContinuedFraction[Khinchin, 30]]] (* G. C. Greubel, May 30 2019 *)

[continued_fraction(khinchin).convergent(n).numerator() for n in (0..30)] # G. C. Greubel, May 30 2019

A127006 Denominators of convergents to Khinchin's constant.

Original entry on oeis.org

1, 1, 3, 16, 19, 35, 89, 124, 213, 763, 7843, 16449, 24292, 89325, 202942, 4959933, 5162875, 20448558, 46059991, 158628531, 204688522, 363317053, 568005575, 51483818803, 103535643181, 155019461984, 1963769186989, 2118788648973

Offset: 1

Eric W. Weisstein, Jan 02 2007

nonn

			2, 3, 8/3, 43/16, 51/19, 94/35, 239/89, 333/124, 572/213, 2049/763, ...

G. C. Greubel, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Khinchin's Constant
Eric Weisstein's World of Mathematics, Khinchin's Constant Digits
Wikipedia, Khinchin's constant

Cf. A002210, A002211, A127005.

Denominator[Convergents[ContinuedFraction[Khinchin, 30]]] (* G. C. Greubel, May 30 2019 *)

[continued_fraction(khinchin).convergent(n).denominator() for n in (0..30)] # G. C. Greubel, May 30 2019

A224851 First position of n in continued fraction for Khinchin's constant.

Original entry on oeis.org

1, 0, 9, 46, 3, 33, 75, 64, 118, 10, 103, 26, 102, 109, 674, 79, 1079, 145, 141, 368, 245, 585, 678, 15, 1427, 1620, 1020, 1626, 63, 1341, 798, 156, 408, 505, 1405, 1782, 1444, 205, 3159, 299, 2682, 2036, 4206, 5203, 270, 522, 367, 7891, 2254, 71, 969, 1493

Offset: 1

Eric W. Weisstein, Jul 22 2013

nonn

Correctly indexed version of A054781.

Smallest positive integers not occurring in the first 106621 terms of the c.f. are 236, 260, 265, 279, 282, 290, 294, 297, 299, ... - Eric W. Weisstein, Oct 01 2011

Eric W. Weisstein, Table of n, a(n) for n = 1..235
Eric Weisstein's World of Mathematics, Khinchin's Constant Continued Fraction.

Cf. A054781(n) (= a(n) + 1).

Cf. A002211 (continued fraction of Khinchin's constant).

a(n) = A054781(n) - 1.

A317907 Number of binary places to which n-th convergent of continued fraction expansion of Khintchine's constant matches the correct value.

Original entry on oeis.org

0, -1, 5, 3, 9, 8, 12, 14, 16, 22, 25, 27, 30, 33, 39, 44, 42, 49, 52, 51, 56, 55, 64, 70, 73, 77, 81, 83, 82, 85, 88, 92, 93, 99, 101, 104, 109, 104, 111, 114, 117, 120, 122, 124, 126, 129, 131, 133, 136, 139, 138, 144, 138, 148, 151, 150, 153, 156, 158, 162

Offset: 1

A.H.M. Smeets, Aug 10 2018

For number of correct decimal digits see A317908.
For the similar case of number of correct binary digits of Pi see A305879.
For the similar case of number of correct binary digits of log(2) see A317557.
The denominator of the k-th convergent obtained from a continued fraction satisfying the Gauss-Kuzmin distribution will tend to exp(k*A100199), A100199 being the inverse of Lévy's constant; the error between the k-th convergent and the constant itself tends to exp(-2*k*A100199), or in binary digits 2*k*A100199/log(2) bits after the binary point.
The sequence for quaternary digits is obtained by floor(a(n)/2), the sequence for octal digits is obtained by floor(a(n)/3), and the sequence for hexadecimal digits is obtained by floor(a(n)/4).

			   n   convergent         binary expansion       a(n)
  ==  =============  ==========================  ====
   1     2 / 1       10.0...                       0
   2     3 / 1       11.0...                      -1
   3     8 / 3       10.101010...                  5
   4    43 / 16      10.1011...                    3
   5    51 / 19      10.1010111100...              9
  oo  lim = A317906  10.101011110111100111...     --

A.H.M. Smeets, Table of n, a(n) for n = 1..19999

Cf. A002210, A002211, A086702, A100199, A305607, A317906, A317908.

i,cf = 0,[]
while i <= 20100:
    c = A002211(i)
    cf,i = cf+[c],i+1
p0,p1,q0,q1,i,base = cf[0],1,1,0,1,2
while i <= 20100:
    p0,p1,q0,q1,i = cf[i]*p0+p1,p0,cf[i]*q0+q1,q0,i+1
a0 = p0//q0
p0 = p0-a0*q0
i,p0,dd = 0,p0*base,[a0]
while i < 70000:
    d,p0,i = p0//q0,(p0%q0)*base,i+1
    dd = dd+[d]
n,pn0,pn1,qn0,qn1 = 1,a0,1,1,0
while n <= 20000:
    p,q = pn0,qn0
    if p//q != a0:
        print(n,"- manual!")
    else:
        i,p,di = 0,(p%q)*base,a0
        while di == dd[i]:
            i,di,p = i+1,p//q,(p%q)*base
        print(n,i-1)
    n,pn0,pn1,qn0,qn1 = n+1,cf[n]*pn0+pn1,pn0,cf[n]*qn0+qn1,qn0

Lim_{n -> oo} a(n)/n = 2*log(A086702)/log(2) = 2*A100199/log(2) = 2*A305607.

A317908 Number of decimal places to which the n-th convergent of the continued fraction expansion of Khintchine's constant matches the correct value.

Original entry on oeis.org

0, -1, 1, 2, 2, 3, 3, 4, 4, 6, 5, 8, 8, 9, 11, 13, 12, 14, 15, 16, 16, 16, 18, 21, 21, 23, 24, 24, 25, 25, 26, 27, 28, 29, 30, 30, 32, 32, 33, 33, 36, 35, 36, 37, 37, 38, 39, 39, 40, 41, 42, 42, 43, 44, 45, 44, 46, 47, 48, 48, 49, 50, 51, 54, 55, 56, 56, 58, 58, 60

Offset: 1

A.H.M. Smeets, Aug 10 2018

Decimal expansion of Khintchine's constant in A002210.
For the similar case of the number of correct decimal digits of Pi see A084407.
For the similar case of the number of correct decimal digits of log(2) see A317558.
For the number of correct binary places see A317907.

			   n   convergent     decimal expansion    a(n)
  ==  =============  ====================  ====
   1     2 / 1       2.0                     0
   2     3 / 1       3.0                    -1
   3     8 / 3       2.66...                 1
   4    43 / 16      2.687...                2
   5    51 / 19      2.684...                2
   6    94 / 35      2.6857...               3
   7   239 / 89      2.6853...               3
   8   333 / 124     2.68548...              4
   9   572 / 213     2.68544...              4
  10  2049 / 763     2.6854521...            6
  oo  lim = A002210  2.685452001065306...   --

A.H.M. Smeets, Table of n, a(n) for n = 1..20000

Cf. A002210, A002211, A086702, A100199, A240995, A317907.

i,cf = 0,[]
while i <= 20100:
    c = A002211(i)
    cf,i = cf+[c],i+1
p0,p1,q0,q1,i,base = cf[0],1,1,0,1,10
while i <= 20100:
    p0,p1,q0,q1,i = cf[i]*p0+p1,p0,cf[i]*q0+q1,q0,i+1
a0 = p0//q0
p0 = p0-a0*q0
i,p0,dd = 0,p0*base,[a0]
while i < 21000:
    d,p0,i = p0//q0,(p0%q0)*base,i+1
    dd = dd+[d]
n,pn0,pn1,qn0,qn1 = 1,a0,1,1,0
while n <= 20000:
    p,q = pn0,qn0
    if p//q != a0:
        print(n,"- manual!")
    else:
        i,p,di = 0,(p%q)*base,a0
        while di == dd[i]:
            i,di,p = i+1,p//q,(p%q)*base
        print(n,i-1)
    n,pn0,pn1,qn0,qn1 = n+1,cf[n]*pn0+pn1,pn0,cf[n]*qn0+qn1,qn0

Limit_{n -> oo} (a(n)/n) = 2*log(A086702)/log(10) = 2*A100199/log(10) = 2*A240995.

cp's OEIS Frontend

A048613 Number of terms (excluding the first) of A002211 for which the geometric mean produces progressively better approximations to Khinchin's constant (itself).

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A002210 Decimal expansion of Khinchin's constant.

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A054781 First position of n in continued fraction for Khinchin's constant.

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A054866 Incrementally largest terms in the continued fraction for Khinchin's constant.

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Mathematica

A054870 Positions of the incrementally largest terms in the continued fraction for Khinchin's constant.

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A127005 Numerators of convergents to Khinchin's constant.

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Sage

A127006 Denominators of convergents to Khinchin's constant.

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Mathematica

Sage

A224851 First position of n in continued fraction for Khinchin's constant.

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A317907 Number of binary places to which n-th convergent of continued fraction expansion of Khintchine's constant matches the correct value.

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