A002210 -id:A002210 - cp's OEIS frontend

A087499 Decimal expansion of Khinchin mean K_{-9}.

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

Offset: 1

Eric W. Weisstein, Sep 09 2003

Khinchin's constant is K_0.

			1.10254313...

Eric Weisstein's World of Mathematics, Khinchin's Constant

Cf. A002210, A087491, A087492, A087493, A087494, A087495, A087496, A087497, A087498, A087499, A087500.

m = 9; digits = 100; exactEnd = 1000; f[n_] = -(Log[1 - (1 + n)^(-2)]/(n^m*Log[2])); s[n_] = Series[f[n], {n, Infinity, digits}] // Normal // N[#, digits]&; exactSum = Sum[f[n], {n, 1, exactEnd}] // N[#, digits]&; extraSum = Sum[s[n], {n, exactEnd + 1, Infinity}] // N[#, digits]&; (exactSum + extraSum)^(-1/m) // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 14 2013 *)

More terms from Jean-François Alcover, Feb 14 2013

A084580 Let y = m/GK(k), where m and k vary over the positive integers and GK(k)=log(1+1/(k*(k+2)))/log(2) is the Gauss-Kuzmin distribution of k. Sort the y values by size and number them consecutively by n. This sequence gives the values of k in order by n.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 4, 2, 1, 3, 1, 2, 1, 5, 1, 1, 2, 1, 3, 6, 1, 4, 2, 1, 1, 1, 2, 3, 1, 7, 1, 2, 1, 5, 1, 4, 2, 1, 3, 1, 8, 1, 2, 1, 1, 3, 2, 1, 6, 1, 4, 9, 1, 2, 1, 5, 1, 3, 2, 1, 1, 1, 2, 10, 1, 4, 3, 1, 7, 2, 1, 1, 1, 2, 1, 3, 5, 1, 11, 2, 6, 1, 4, 1, 2, 1, 3, 1, 1, 8, 2, 1, 1, 12, 2, 1, 3, 4, 1, 1

Offset: 1

Paul D. Hanna, May 31 2003

nonn

The geometric mean of the sequence equals Khintchine's constant K=2.685452001 = A002210 since the frequency of the integers agrees with the Gauss-Kuzmin distribution. When considered as a continued fraction, the resulting constant is 0.5815803358828329856145... = A372869 = [0;1,1,2,1,1,3,2,1,1,1,4,2,1,...].

This can also be defined as the sequence formed by greedily sampling the Gauss-Kuzmin distribution. - Jwalin Bhatt, Nov 28 2024

			From _Jwalin Bhatt_, Jul 25 2025: (Start)
Constructing the sequence by greedily sampling the Gauss-Kuzmin distribution to minimize discrepancy.
Let p(n) denote the probability of n and c(n) denote the count of occurrences of n so far.
We take the ratio of the actual occurrences c(n)+1 to the probability and pick the one with the lowest value.
Since p(n) is monotonic decreasing, we only need to compute c(n) once we see c(n-1).
  | n | (c(1)+1)/p(1) | (c(2)+1)/p(2) | (c(3)+1)/p(3) | choice |
  |---|---------------|---------------|---------------|--------|
  | 1 |     5.884     |       -       |       -       |   1    |
  | 2 |     4.818     |     5.884     |       -       |   1    |
  | 3 |     7.228     |     5.884     |       -       |   2    |
  | 4 |     7.228     |    11.769     |    10.740     |   1    |
  | 5 |     9.637     |    11.769     |    10.740     |   1    |
  | 6 |    12.047     |    11.769     |    10.740     |   3    | (End)

Jwalin Bhatt, Table of n, a(n) for n = 1..10000
Wikipedia, Gauss Kuzmin distribution

Cf. A002210, A084576-A084579, A084581-A084587, A372869, A386016.

pdf[k_] := Log[1 + 1/(k*(k + 2))]/Log[2]
samplePDF[numCoeffs_] := Module[
  {coeffs = {}, counts = {0}, minTime, minIndex, time},
Do[
    minTime = Infinity;
    Do[
      time = (counts[[i]] + 1)/pdf[i];
      If[time < minTime, minIndex = i; minTime = time],
      {i, 1, Length[counts]}
    ];
    If[minIndex == Length[counts], AppendTo[counts, 0]];
    counts[[minIndex]] += 1;
    AppendTo[coeffs, minIndex],
    {numCoeffs}
  ];
  coeffs
]
A084580 = samplePDF[120]  (* Jwalin Bhatt, Jul 25 2025 *)

import math
def sample_gauss_kuzmin_distribution(num_coeffs):
  coeffs, counts = [], [0]
  for _ in range(num_coeffs):
    min_time = math.inf
    for i, count in enumerate(counts, start=1):
      time = (count+1) / -math.log2(1-(1/((i+1)**2)))
      if time < min_time:
        min_index, min_time = i, time
    if min_index == len(counts):
      counts.append(0)
    counts[min_index-1] += 1
    coeffs.append(min_index)
  return coeffs
A084580 = sample_gauss_kuzmin_distribution(100) # Jwalin Bhatt, Dec 22 2024

A244109 Decimal expansion of a partial sum limiting constant related to the Lüroth representation of real numbers.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 20 2014

			2.04627745285587859107017615395043619498429055873216651873269723582433...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 1.8.1 Alternative representations [of real numbers], p. 62.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 538.
Sofia Kalpazidou, Khintchine's constant for Lüroth representation, Journal of Number Theory, Volume 29, Issue 2, June 1988, Pages 196-205.

Cf. A002210, A085361. Equals twice A340440.

SetDefaultRealField(RealField(120)); L:=RiemannZeta(); (&+[((1-(-1)^n)*Evaluate(L,n+1)-1)/n: n in [1..1000]]); // G. C. Greubel, Nov 15 2018

evalf(Sum(((1 + (-1)^(n+1))*Zeta(n+1) - 1)/n, n=1..infinity), 120); # Vaclav Kotesovec, Dec 11 2015

NSum[Log[k*(k+1)]/(k*(k+1)), {k, 1, Infinity}, WorkingPrecision -> 120, NSumTerms -> 5000, Method -> {NIntegrate, MaxRecursion -> 100}] (* Vaclav Kotesovec, Dec 11 2015 *)
digits = 120; RealDigits[NSum[((1-(-1)^n)*Zeta[n+1] -1)/n, {n, 1, Infinity}, NSumTerms -> 20*digits, WorkingPrecision -> 10*digits, Method -> "AlternatingSigns"], 10, digits][[1]] (* G. C. Greubel, Nov 15 2018 *)

default(realprecision, 1000); s = sumalt(n=1, ((1 + (-1)^(n+1))*zeta(n+1) - 1)/n); default(realprecision, 100); print(s) \\ Vaclav Kotesovec, Dec 11 2015

2*suminf(k=1, -zeta'(2*k)) \\ Vaclav Kotesovec, Jun 17 2021

numerical_approx(sum(((1-(-1)^k)*zeta(k+1)-1)/k for k in [1..1000]), digits=120) # G. C. Greubel, Nov 15 2018

Equals Sum_{k>=1} log(k*(k+1))/(k*(k+1)).
Equals A085361 + A131688. - Vaclav Kotesovec, Dec 11 2015
Equals Sum_{n >=1} ((1 + (-1)^(n+1))*zeta(n + 1) - 1)/n. - G. C. Greubel, Nov 15 2018
Equals 2*Sum_{k>=2} log(k)/(k^2-1) = 2*A340440. - Gleb Koloskov, May 02 2021
Equals -2*Sum_{k>=1} zeta'(2*k). - Vaclav Kotesovec, Jun 17 2021

Corrected by Vaclav Kotesovec, Dec 11 2015

A245254 Decimal expansion of U = Product_{k>=1} (k^(1/(k*(k+1)))), a Khintchine-like limiting constant related to Lüroth's representation of real numbers.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jul 15 2014

The geometric mean of the Yule-Simon distribution with parameter value 1 (A383855) approaches this constant. In general, the geometric mean of the Yule-Simon distribution approaches Product_{k>=2} k^(1/(p*Beta(k,p+1))). - Jwalin Bhatt, May 12 2025

			2.200161058099026553194557866559944872685662324752723888723145117763169...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 1.8.1 Alternative representations [of real numbers], p. 62.

Sofia Kalpazidou, Khintchine's constant for Lüroth representation, Journal of Number Theory, Vol. 29, No. 2 (June 1988), pp. 196-205.

Cf. A002210, A085291, A085361, A242624, A244109(V), A131688(W), A383855.

evalf(exp(Sum((Zeta(n+1)-1)/n, n=1..infinity)), 120); # Vaclav Kotesovec, Dec 11 2015

Exp[NSum[Log[k]/(k*(k+1)), {k, 1, Infinity}, WorkingPrecision -> 120, NSumTerms -> 5000, Method -> {NIntegrate, MaxRecursion -> 100}]] (* Vaclav Kotesovec, Dec 11 2015 *)

Equals exp(A085361).
U*V*W = 1, where V is A244109 and W is A131688.
Equals e * A085291. - Amiram Eldar, Jun 27 2021
Equals 1/A242624. - Amiram Eldar, Feb 06 2022

Corrected by Vaclav Kotesovec, Dec 11 2015

A245255 Decimal expansion of y_1, the first of four non-explicit constants recursively derived from Khintchine's [Khinchin's] constant.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jul 15 2014

			2.303842196283770422112375608882267846971196077828808534219305173...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 1.8.1 Alternative representations [of real numbers], p. 63.

Herman P. Robinson, Mathematical Constants.

Cf. A002210, A245256, A245257, A245258.

K = Khinchin; a[0] = pi[0] = 2; pi[n_] := Product[a[i], {i, 0, n}]; Clear[a]; a[n_] := a[n] = Floor[K^(n+1)/pi[n-1]]; FromContinuedFraction[Array[a, 300, 0]] // RealDigits[#, 10, 105]& // First

A245256 Decimal expansion of y_2, the second of four non-explicit constants recursively derived from Khintchine's [Khinchin's] constant.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jul 15 2014

			3.30384219630718251298905725146305146363000806852201418586337176944971...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 1.8.1 Alternative representations [of real numbers], p. 63.

Herman P. Robinson, Mathematical Constants.

Cf. A002210, A245255, A245257, A245258.

K = Khinchin; a[0] = pi[0] = 3; pi[n_] := Product[a[i], {i, 0, n}]; Clear[a]; a[n_] := a[n] = Floor[K^(n+1)/pi[n-1]] + 1; FromContinuedFraction[Array[a, 300, 0]] // RealDigits[#, 10, 102]& // First

A245257 Decimal expansion of y_3, the third of four non-explicit constants recursively derived from Khintchine's [Khinchin's] constant.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jul 15 2014

			2.224751480980583015375598924924190424236366707982466701694563315747...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 1.8.1 Alternative representations [of real numbers], p. 63.

Herman P. Robinson, Mathematical Constants.

Cf. A002210, A245255, A245256, A245258.

K = Khinchin; a[0] = pi[0] = 2; pi[n_] := Product[a[i], {i, 0, n}]; Clear[a]; a[n_?EvenQ] := a[n] = Floor[K^(n+1)/pi[n-1]]; a[n_?OddQ] := a[n] = Ceiling[K^(n+1)/pi[n-1]]; FromContinuedFraction[Array[a, 300, 0]] // RealDigits[#, 10, 105]& // First

A245258 Decimal expansion of y_4, the last of four non-explicit constants recursively derived from Khintchine's [Khinchin's] constant.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jul 15 2014

			3.4493588902597404159513218512538883603456245038254159108814941005755696...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 1.8.1 Alternative representations [of real numbers], p. 63.

Herman P. Robinson, Mathematical Constants.

Cf. A002210, A245255, A245256, A245257.

K = Khinchin; a[0] = pi[0] = 3; pi[n_] := Product[a[i], {i, 0, n}]; Clear[a]; a[n_?EvenQ] := a[n] = Ceiling[K^(n+1)/pi[n-1]]; a[n_?OddQ] := a[n] = Floor[K^(n+1)/pi[n-1]]; FromContinuedFraction[Array[a, 300, 0]] // RealDigits[#, 10, 105]& // First

A089618 Continued fraction elements constructed out of a van der Corput discrepancy sequence. Interpreted as such, it is the simple continued fraction of 0.461070495956719519354149869336699687678...

Original entry on oeis.org

0, 2, 5, 1, 11, 1, 3, 1, 22, 2, 4, 1, 7, 1, 2, 1, 45, 2, 4, 1, 8, 1, 3, 1, 14, 1, 3, 1, 6, 1, 2, 1, 91, 2, 4, 1, 9, 1, 3, 1, 17, 2, 3, 1, 6, 1, 2, 1, 30, 2, 4, 1, 7, 1, 2, 1, 12, 1, 3, 1, 5, 1, 2, 1, 184, 2, 5, 1, 10, 1, 3, 1, 20, 2, 4, 1, 6, 1, 2, 1, 36, 2, 4

Offset: 0

Hans Havermann, Jan 03 2004

The authors of On the Khintchine Constant posit that the geometric mean of the sequence (interpreted as a simple continued fraction expansion) is Khinchin's constant "on the idea that the discrepancy sequence is in a certain sense equidistributed."

That conjecture has been proven by Wieting. Moreover, the r-th power mean of the sequence (except a(0)=0, of course) also converges to the corresponding constant K_r for any real r<1. - Andrey Zabolotskiy, Feb 20 2017

			40 is 101000 in base 2, so b(40) = 0.078125 (the equivalent of binary 0.000101), 1/(2^0.078125-1) is approximately 17.97 and a(40) is the integer part of this: 17.

Andrey Zabolotskiy, Table of n, a(n) for n = 0..10000
D. Bailey, J. Borwein, & R. Crandall, On the Khintchine constant, Mathematics of Computation 66:217 (January 1997), pp. 417-431.
T. Wieting, A Khinchin Sequence, Proc. Amer. Math. Soc., 136 (2008), 815-824.

Cf. A002210, A087491-A087500, A030101.

a[n_] := (m = IntegerDigits[n, 2]; l = Length[m]; s = "2^^."; Do[s = s <> ToString[m[[i]]], {i, l, 1, -1}]; Floor[1/(2^ToExpression[s]-1)]); Prepend[Table[a[i], {i, 1, 120}], 0]
a[n_] := If[n==0, 0, Floor[1 / (2^FromDigits[{Reverse[IntegerDigits[n,2]],0},2] - 1)]]; (* Andrey Zabolotskiy, Feb 20 2017 *)

a(n) = integer part of 1/(2^b(n)-1) where b(n) = digit-reversal of binary of (positive integer) n, preceded by a decimal point and converted (from base 2) to base 10; initial term, a(0), is defined as 0.

a(n) = floor(1/(2^(A030101(n)/A062383(n))-1)) for n>0. - Andrey Zabolotskiy, Feb 20 2017

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

cp's OEIS Frontend

A087499 Decimal expansion of Khinchin mean K_{-9}.

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A084580 Let y = m/GK(k), where m and k vary over the positive integers and GK(k)=log(1+1/(k*(k+2)))/log(2) is the Gauss-Kuzmin distribution of k. Sort the y values by size and number them consecutively by n. This sequence gives the values of k in order by n.

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A244109 Decimal expansion of a partial sum limiting constant related to the Lüroth representation of real numbers.

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A245254 Decimal expansion of U = Product_{k>=1} (k^(1/(k*(k+1)))), a Khintchine-like limiting constant related to Lüroth's representation of real numbers.

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A245255 Decimal expansion of y_1, the first of four non-explicit constants recursively derived from Khintchine's [Khinchin's] constant.

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A245256 Decimal expansion of y_2, the second of four non-explicit constants recursively derived from Khintchine's [Khinchin's] constant.

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A245257 Decimal expansion of y_3, the third of four non-explicit constants recursively derived from Khintchine's [Khinchin's] constant.

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