A086702 -id:A086702 - cp's OEIS frontend

A120028 Continued fraction expansion of the value of Minkowski's question mark function at Levy's constant (Exp[Pi^2/(12*Log[2])], A086702).

3, 6, 10, 2, 8388607, 1, 1, 10, 5, 1, 19802808501020211999048785114, 1, 5, 1, 5, 2, 2, 10, 4, 1, 142, 3, 1, 1, 1, 1, 1, 2, 13, 16, 1, 83, 2, 4, 1, 15, 1, 62, 1, 20, 1, 2, 1, 1, 1, 9, 1, 1, 1, 13, 2, 3, 1, 4, 1, 5, 1, 1, 1, 5, 7, 1, 27, 1, 2, 4, 3, 1, 3, 3, 1, 7, 2, 1, 1, 91, 11, 1, 2, 4, 4

Offset: 0

Joseph Biberstine (jrbibers(AT)indiana.edu), Jun 04 2006

a[92] has over 150 decimal digits, making 750332738256083509758042341909438953923620244270237443771885409340366143805720089/2^267 an excellent approximation to the constant.

Cf. A120029.

ContinuedFraction[cf = ContinuedFraction[Exp[Pi^2/(12*Log[2])], 50(*arbitrary precision*)]; IntegerPart[Exp[Pi^2/(12*Log[2])]] + Sum[(-1)^(k)/2^(Sum[cf[[i]], {i, 2, k}] - 1), {k, 2, Length[cf]}]]

A120029 Decimal expansion of the value of Minkowski's question mark function at Levy's constant (Exp[Pi^2/(12*Log[2])], A086702).

Original entry on oeis.org

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

Offset: 1

Joseph Biberstine (jrbibers(AT)indiana.edu), Jun 04 2006

a[92] has over 150 decimal digits, making 750332738256083509758042341909438953923620244270237443771885409340366143805720089/2^267 an excellent approximation to the constant.

			3.164062499992724042385816574096679687499999999999997...

Index entries for sequences related to Minkowski's question mark function

Cf. A120028.

RealDigits[cf = ContinuedFraction[Exp[Pi^2/(12*Log[2])], 50(*arbitrary precision*)]; IntegerPart[Exp[Pi^2/(12*Log[2])]] + Sum[(-1)^(k)/2^(Sum[cf[[i]], {i, 2, k}] - 1), {k, 2, Length[cf]}], 10, 100]

A100199 Decimal expansion of Pi^2/(12*log(2)), inverse of Levy's constant.

Original entry on oeis.org

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

Offset: 1

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Dec 27 2004

From A.H.M. Smeets, Jun 12 2018: (Start)
The denominator of the k-th convergent obtained from a continued fraction of a constant, the terms of the continued fraction satisfying the Gauss-Kuzmin distribution, will tend to exp(k*A100199).
Similarly, the error between the k-th convergent obtained from a continued fraction of a constant, and the constant itself will tend to exp(-2*k*A100199). (End)
The term "Lévy's constant" is sometimes used to refer to this constant (Wikipedia). - Bernard Schott, Sep 01 2022

			1.1865691104156254528217229759472371205683565364720543359542542986528...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.7, p. 54.

G. C. Greubel, Table of n, a(n) for n = 1..10000
R. M. Corless, Continued Fractions and Chaos, Amer. Math. Monthly 99, 203-215, 1992.
Eric Weisstein's World of Mathematics, Khinchin-Levy Constant.
Eric Weisstein's World of Mathematics, Lévy Constant.
Wikipedia, Lévy's constant.

Cf. A086702, A089729, A072691, A002162, A174606.

RealDigits[Pi^2/(12*Log[2]), 10, 100][[1]] (* G. C. Greubel, Mar 23 2017 *)

Pi^2/log(4096) \\ Charles R Greathouse IV, Aug 04 2016

Equals 1/A089729 = log(A086702) = A174606/2.
Equals ((Pi^2)/12)/log(2) = A072691 / A002162 = (Sum_{n>=1} ((-1)^(n+1))/n^2) / (Sum_{n>=1} ((-1)^(n+1))/n^1). - Terry D. Grant, Aug 03 2016
Equals (-1/log(2)) * Integral_{x=0..1} log(x)/(1+x) dx (from Corless, 1992). - Bernard Schott, Sep 01 2022

A084407 Number of decimal places to which the n-th convergent of continued fraction expansion of Pi matches with the correct value.

Original entry on oeis.org

0, 2, 4, 6, 9, 9, 9, 9, 11, 10, 12, 12, 14, 15, 15, 16, 17, 17, 18, 19, 21, 23, 24, 24, 25, 27, 29, 30, 30, 32, 33, 34, 37, 39, 40, 40, 41, 42, 44, 45, 45, 46, 47, 49, 50, 51, 51, 53, 54, 55, 55, 56, 56, 58, 59, 59, 60, 60, 61, 60, 62, 64, 63, 64, 65, 65, 67, 67, 68, 70, 69, 71

Offset: 1

Lekraj Beedassy, Jun 24 2003

The n-th convergent of the continued fraction expansion of Pi is A002485(n+1)/A002486(n+1).

			From _A.H.M. Smeets_, Jun 13 2018: (Start)
Pi = 3.141592653589...
n=1: 3/1 = 3.0... so a(1) = 0;
n=2: 22/7 = 3.142... so a(2) = 2;
n=3: 333/106 = 3.14150... so a(3) = 4;
n=4: 355/113 = 3.1415929... so a(4) = 6;
n=5: 103993/33102 = 3.1415926530... so a(5) = 9;
n=6: 104348/33215 = 3.1415926539... so a(6) = 9;
n=7: 208341/66317 = 3.1415926534... so a(7) = 9;
n=8: 312689/99532 = 3.1415926536... so a(8) = 9;
n=9: 833719/265381 = 3.141592653581... so a(9) = 11;
n=10: 1146408/364913 = 3.14159265359... so a(10) = 10. (End)

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

Cf. A000796.
Cf. A002485, A002486, A114526.
Cf. A086702, A100199, A240995.

Limit_{n -> oo} a(n)/n = 2*log(A086702)/log(10) = 2*A100199/log(10) = 2*A240995. - A.H.M. Smeets, Jun 13 2018

More terms from Vladeta Jovovic, Jun 27 2003

A089729 Decimal expansion of Levy's constant 12*log(2)/Pi^2.

Original entry on oeis.org

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

Offset: 0

Benoit Cloitre, Jan 19 2004

For x>y in [1..n], the average number of loop steps of the Euclid Algorithm for GCD (over all choices x, y) is asymptotic to k*log(n) where k is this constant. See Crandall & Pomerance. - Michel Marcus, Mar 23 2016

			0.8427659132721945169072631939639641155945183893191504965...

R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see Theorem 2.1.3, p. 84.
S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 156.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Harmonic Number
Eric Weisstein's World of Mathematics, Levy Constant

Cf. A086702, A086237.

RealDigits[12 Log[2]/Pi^2, 10, 100][[1]] (* Bruno Berselli, Jun 20 2013 *)

12*log(2)/Pi^2 \\ Michel Marcus, Mar 23 2016

Leading zero removed by R. J. Mathar, Feb 05 2009

A305879 Number of binary places to which n-th convergent of continued fraction expansion of Pi matches the correct value.

Original entry on oeis.org

2, 8, 13, 21, 28, 31, 28, 34, 32, 38, 40, 44, 47, 51, 52, 54, 57, 60, 62, 64, 70, 78, 80, 81, 84, 91, 94, 100, 103, 104, 107, 116, 121, 132, 133, 136, 133, 144, 148, 152, 148, 156, 158, 165, 167, 170, 173, 176, 179, 182

Offset: 1

A.H.M. Smeets, Jun 13 2018

For the similar case of number of correct decimal places see A084407.
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), the sequence for hexadecimal digits is obtained by floor(a(n)/4).

			Pi = 11.0010010000111111...
n=1: 3/1 = 11.000... so a(1) = 2
n=2: 22/7 = 11.001001001... so a(2) = 8
n=3: 333/106 = 11.00100100001110... so a(3) = 13

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

Cf. A084407, A086702, A100199, A305607.

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

A317557 Number of binary digits to which the n-th convergent of the continued fraction expansion of log(2) matches the correct value.

Original entry on oeis.org

0, -1, 3, 6, 9, 13, 14, 17, 19, 20, 23, 20, 25, 20, 33, 37, 35, 38, 41, 43, 45, 43, 47, 48, 52, 54, 58, 61, 68, 70, 74, 77, 78, 81, 86, 89, 92, 93, 92, 99, 105, 109, 113, 116, 118, 121, 127, 133, 136, 135, 139, 141, 145, 149, 154, 159, 161, 165, 171, 173, 172, 180

Offset: 1

A.H.M. Smeets, Jul 31 2018

Binary expansion of log(2) in A068426.
For number of correct decimal digits see A317558.
For the similar case of number of correct binary digits of Pi see A305879.
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     0 / 1      0.0                              0
   2     1 / 1      1.0                             -1
   3     2 / 3      0.1010...                        3
   4     7 / 10     0.1011001...                     6
   5     9 / 13     0.1011000100...                  9
   6    61 / 88     0.10110001011101...             13
   7   192 / 277    0.101100010111000...            14
   8   253 / 365    0.101100010111001001...         17
   9   445 / 642    0.10110001011100100000...       19
  10  1143 / 1649   0.101100010111001000011...      20
  oo  lim = log(2)  0.101100010111001000010111...   --

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

Cf. A016730, A068426, A086702, A100199, A305607, A317558.

a[n_] := Block[{k = 1, a = RealDigits[ Log@2, 2, 4 + 10][[1]], b = RealDigits[ FromContinuedFraction@ ContinuedFraction[Log@2, n + 1], 2, 4n + 10][[1]]}, While[ a[[k]] == b[[k]], k++]; k - 1]; a[1] = 0; a[2] = -1; Array[a, 61] (* Robert G. Wilson v, Aug 09 2018 *)

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

a(40) onward from Robert G. Wilson v, Aug 09 2018

A317558 Number of decimal digits to which the n-th convergent of the continued fraction expansion of log(2) matches the correct value.

Original entry on oeis.org

0, -1, 1, 0, 2, 4, 5, 4, 5, 6, 6, 6, 7, 8, 9, 10, 11, 10, 12, 13, 13, 13, 14, 15, 15, 16, 17, 18, 20, 22, 22, 23, 23, 24, 25, 26, 27, 27, 28, 29, 31, 32, 33, 34, 35, 36, 38, 40, 39, 41, 39, 43, 44, 45, 46, 48, 48, 49, 51, 52, 52, 54, 54, 55, 55, 56, 57, 57, 58

Offset: 1

A.H.M. Smeets, Jul 31 2018

Decimal expansion of log(2) in A002162.
For the number of correct binary digits see A317557.
For the similar case of number of correct decimal digits of Pi see A084407.

			   n   convergent    decimal expansion    a(n)
  ==  ============  ====================  ====
   1     0 / 1      0.0                     0
   2     1 / 1      1.0                    -1
   3     2 / 3      0.66...                 1
   4     7 / 10     0.7...                  0
   5     9 / 13     0.692...                2
   6    61 / 88     0.69318...              4
   7   192 / 277    0.693140...             5
   8   253 / 365    0.69315...              4
   9   445 / 642    0.693146...             5
  10  1143 / 1649   0.6931473...            6
  oo  lim = log(2)  0.693147180559945...   --

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

Cf. A002162, A016730, A086702, A100199, A240995, A317557.

a[n_] := Block[{k = 1, a = RealDigits[Log@2, 10, n + 10][[1]], b = RealDigits[ FromContinuedFraction@ ContinuedFraction[ Log@2, n], 10, n + 10][[1]]}, While[a[[k]] == b[[k]], k++]; k - 1]; a[1] = 0; a[2] = -1; Array[a, 69] (* Robert G. Wilson v, Aug 09 2018 *)

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

a(61) onward from Robert G. Wilson v, Aug 09 2018

A174606 Decimal expansion of Pi^2/(6*log 2).

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, Mar 23 2010

			2.37313822083125090564344595....

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.8.1, p. 62.

Brigitte Vallée, Digits and continuants in Euclidean algorithms. Ergodic vs tauberian theorems, Journal de théorie des nombres de Bordeaux 12 (2000), 519-558.

Cf. A002162, A013661, A086702, A100199.

RealDigits[Pi^2/(6*Log[2]), 10, 100][[1]] (* Amiram Eldar, May 24 2023 *)

Pi^2/6/log(2) \\ Michel Marcus, Apr 04 2014

Equals 2*log(A086702). - Jean-François Alcover, Apr 15 2014

Equals 2*A100199. - Hugo Pfoertner, Nov 18 2024

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.

cp's OEIS Frontend

A120028 Continued fraction expansion of the value of Minkowski's question mark function at Levy's constant (Exp[Pi^2/(12*Log[2])], A086702).

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A120029 Decimal expansion of the value of Minkowski's question mark function at Levy's constant (Exp[Pi^2/(12*Log[2])], A086702).

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A100199 Decimal expansion of Pi^2/(12*log(2)), inverse of Levy's constant.

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A084407 Number of decimal places to which the n-th convergent of continued fraction expansion of Pi matches with the correct value.

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A089729 Decimal expansion of Levy's constant 12*log(2)/Pi^2.

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A305879 Number of binary places to which n-th convergent of continued fraction expansion of Pi matches the correct value.

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A317557 Number of binary digits to which the n-th convergent of the continued fraction expansion of log(2) matches the correct value.

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A317558 Number of decimal digits to which the n-th convergent of the continued fraction expansion of log(2) matches the correct value.

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