A089729 -id:A089729 - cp's OEIS frontend

A055573 Number of terms in simple continued fraction for n-th harmonic number H_n = Sum_{k=1..n} (1/k).

1, 2, 3, 2, 5, 4, 6, 7, 10, 8, 7, 10, 15, 9, 9, 17, 18, 11, 20, 16, 18, 18, 23, 19, 24, 25, 24, 26, 29, 21, 24, 23, 26, 25, 32, 34, 33, 26, 24, 31, 32, 31, 36, 36, 39, 32, 34, 42, 47, 44, 46, 35, 40, 48, 43, 47, 59, 50, 49, 39, 50, 66, 54, 44, 54, 49, 41, 64, 47, 46, 54, 71, 72

Offset: 1

Leroy Quet, Jul 10 2000

nonn

By "simple continued fraction" is meant a continued fraction whose terms are positive integers and the final term is >= 2.

Does any number appear infinitely often in this sequence?

			Sum_{k=1 to 3} [1/k] = 11/6 = 1 + 1/(1 + 1/5), so the 3rd term is 3 because the simple continued fraction for the 3rd harmonic number has 3 terms.

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 156

G. C. Greubel, Table of n, a(n) for n = 1..10000 (terms 1..500 from M. F. Hasler)
Eric Weisstein's World of Mathematics, Harmonic Number
Eric Weisstein's World of Mathematics, Continued Fraction
G. Xiao, Contfrac server, To evaluate H(m) and display its continued fraction expansion, operate on "sum(n=1, m, 1/n)"

m-th harmonic number H(m) = A001008(m)/A002805(m).
Cf. A058027, A100398, A110020, A112286, A112287.
Cf. A139001 (partial sums).

Table[ Length[ ContinuedFraction[ HarmonicNumber[n]]], {n, 1, 75}] (* Robert G. Wilson v, Dec 22 2003 *)

c=0;h=0;for(n=1,500,write("projects/b055573.txt",c++," ",#contfrac(h+=1/n))) \\ M. F. Hasler, May 31 2008

from sympy import harmonic
from sympy.ntheory.continued_fraction import continued_fraction
def A055573(n): return len(continued_fraction(harmonic(n))) # Chai Wah Wu, Jun 27 2024

It appears that lim n -> infinity a(n)/n = C = 0.84... - Benoit Cloitre, May 04 2002

Conjecture: limit n -> infinity a(n)/n = 12*log(2)/Pi^2 = 0.84..... = A089729 Levy's constant. - Benoit Cloitre, Jan 17 2004

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

A240995 Decimal expansion of Pi^2/(12log(2)log(10)), a constant appearing in several contexts, namely, Khintchine-Lévy Constants, Gauss-Kuzmin distribution and Pell's equation.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Aug 07 2014

			0.515320417050356467940888047...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Steven R. Finch, Errata and Addenda to Mathematical Constants, sections 1.8 and 2.17
T. Oliveira e Silva, Large fundamental solutions of Pell's equation
Eric Weisstein's MathWorld, Kuzmin-Wirsing Constant

Cf. A038517, A086819, A089729.

RealDigits[Pi^2/(12*Log[2]*Log[10]), 10, 103] // First

Pi^2/(12*log(2)*log(10)) \\ G. C. Greubel, Mar 23 2017

Equals 1/(2*A086819).

Also equals 1/(log(10)*A089729).

A090397 Number of terms in simple continued fraction for (3/2)^n.

Original entry on oeis.org

1, 2, 2, 4, 2, 5, 7, 6, 10, 8, 8, 5, 8, 13, 7, 10, 13, 10, 11, 10, 14, 19, 15, 13, 12, 13, 16, 17, 15, 11, 17, 24, 21, 29, 23, 18, 23, 24, 26, 31, 28, 26, 29, 31, 24, 31, 28, 26, 20, 18, 18, 23, 30, 33, 36, 36, 31, 33, 38, 37, 35, 37, 29, 40, 43, 45, 39, 37, 40, 39, 42, 47, 41, 33

Offset: 0

Benoit Cloitre, Jan 31 2004

nonn

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 156.

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A089729.

Table[Length[ContinuedFraction[(3/2)^n]], {n,0,100}] (* G. C. Greubel, Mar 23 2017 *)

PARI
```
a(n)=length(contfrac((3/2)^n))
```

Conjecture : a(n) is asymptotic to c*n where c=12*(log(2)/Pi)^2=log(2)*Levy's constant=0.58416081665664902187922697....

A275696 Decimal expansion of e^(3Zeta(3)/(4log(2))).

Original entry on oeis.org

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

Offset: 1

Terry D. Grant, Aug 05 2016

			3.67168670743006450078...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A275689, A086702, A089729, A100199.

RealDigits[Exp[3*(Zeta[3]/(4*Log[2]))],10,100]

exp(3*(zeta(3)/(4*log(2)))) \\ G. C. Greubel, Mar 23 2017

Equals e^(3*Zeta(3)/(4*log(2))) = A001113^(A275689).

A375066 Decimal expansion of Hensley's constant, arising in the analysis of the Euclidean algorithm.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Jul 29 2024

Appears in the formula for the asymptotic variance of the Euclidean algorithm.

When applying the Euclidean algorithm on pairs (a, b), with 0 <= a <= b <= x, the asymptotic formula for the variance of the number of steps (divisions), as x -> infinity, is H*log(x), where H is this constant. See Lhote (2004), eq. 1.8.

			0.51606240889999180681...

Philippe Flajolet and Brigitte Vallée, Continued Fractions, Comparison Algorithms, and Fine Structure Constants, Research Report RR-4072, INRIA, 2000, p. 24.
Doug Hensley, The Number of Steps in the Euclidean Algorithm, Journal of Number Theory, Vol. 49, Issue 2, November 1994, pp. 142-182.
Doug Hensley, Continued Fractions, World Scientific, 2006, p. 210.
Loïck Lhote, Computation of a class of continued fraction constants, in Arge et al., Proceedings of the Sixth Workshop on Algorithm Engineering and Experiments and the First Workshop on Analytic Algorithmics and Combinatorics, 2004, pp. 199-210.
Wikipedia, Euclidean algorithm.

Cf. A089729, A174606, A345987, A362149, A362150.

Equals 2*(lambda''(1) - lambda'(1)^2) / (-lambda'(1)^3), where lambda'(1) = -Pi^2/(6*log(2)) = -A174606 and lambda''(1) is 9.08037... See Lhote (2004), eq. 1.8, and Flajolet and Vallée (2000), p. 24 (where lambda''(1) is called the Hensley's constant).

cp's OEIS Frontend

A055573 Number of terms in simple continued fraction for n-th harmonic number H_n = Sum_{k=1..n} (1/k).

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

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A240995 Decimal expansion of Pi^2/(12*log(2)*log(10)), a constant appearing in several contexts, namely, Khintchine-Lévy Constants, Gauss-Kuzmin distribution and Pell's equation.

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A090397 Number of terms in simple continued fraction for (3/2)^n.

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Keywords

References

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Programs

Mathematica

PARI

Formula

A275696 Decimal expansion of e^(3*Zeta(3)/(4*log(2))).

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Examples

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Programs

Mathematica

PARI

Formula

A375066 Decimal expansion of Hensley's constant, arising in the analysis of the Euclidean algorithm.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A240995 Decimal expansion of Pi^2/(12log(2)log(10)), a constant appearing in several contexts, namely, Khintchine-Lévy Constants, Gauss-Kuzmin distribution and Pell's equation.

A275696 Decimal expansion of e^(3Zeta(3)/(4log(2))).