A102525 -id:A102525 - cp's OEIS frontend

A355512 Sum of numerator and denominator in the convergents of the approximation of log(2)/log(3) by a continued fraction.

2, 3, 5, 13, 31, 106, 137, 791, 1719, 40328, 82375, 205078, 287453, 492531, 27376658, 27869189, 138853414, 444429431, 583282845, 1027712276, 15998966985, 17026679261, 169239080334, 355504839929, 1946763279979, 13982847799782, 15929611079761, 29912458879543, 135579446597933

Offset: 1

Hugo Pfoertner, Jul 05 2022

nonn

Cf. A005663, A005664, A102525, A355513, A355515.

Cf. A355514 for the relation to potential cycle lengths of the 3x+1 problem.

a355512(upto) = {my(q=log(2)/log(3), fa=oo); for (denmax=1, upto, my(f=bestappr(q,denmax)); if (fa!=f, print1(numerator(f)+denominator(f),", "); fa=f))};
\\ needs increased precision for larger terms
a355512(10^6)

\\ See also A005663 and A005664 for efficient code.

A355513 Sum of numerator and denominator in a rational approximation j/k of q = log(2)/log(3), such that abs(j/k - q) is a new minimum.

Original entry on oeis.org

2, 3, 5, 8, 13, 18, 31, 75, 106, 137, 517, 654, 791, 928, 1719, 21419, 23138, 24857, 26576, 28295, 30014, 31733, 33452, 35171, 36890, 38609, 40328, 82375, 205078, 287453, 492531, 14078321, 14570852, 15063383, 15555914, 16048445, 16540976, 17033507, 17526038, 18018569

Offset: 1

Hugo Pfoertner, Jul 05 2022

nonn

Cf. A102525, A355512, A355514, A355515.

a355513(upto) = {my(q=log(2)/log(3), dmin=oo);for (m=1, upto, my(n=round(m*q), qq=n/m, d=abs(qq-q)); if(d

A355515 Sum of numerator and denominator in a rational approximation j/k of q = log(2)/log(3), such that j/k - q is a new minimum, i.e., q is approximated from above.

Original entry on oeis.org

2, 5, 18, 31, 137, 928, 1719, 42047, 82375, 287453, 779984, 1272515, 1765046, 2257577, 2750108, 3242639, 3735170, 4227701, 4720232, 5212763, 5705294, 6197825, 6690356, 7182887, 7675418, 8167949, 8660480, 9153011, 9645542, 10138073, 10630604, 11123135, 11615666, 12108197

Offset: 1

Hugo Pfoertner, Jul 05 2022

nonn

Cf. A102525, A355512, A355513, A355514.

a355515(upto) = {my(q=log(2)/log(3), dmin=oo); for (m=1, upto, my(n=ceil(m*q), qq=n/m, d=qq-q); if (d

A152549 Decimal expansion of log_3(18).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Oct 28 2009, based on a posting by Michael Kleber to the Math Fun Mailing List

Hausdorff dimension of Jeannine Mosely's origami model of Menger's sponge.

			2.6309297535714574370995271143427608542995856401318804278706...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Seb Perez-D, Illustration of sponge (1)
Seb Perez-D, Illustration of sponge (2)
Seb Perez-D, Illustration of sponge (3)
Index entries for transcendental numbers

RealDigits[Log[3, 18], 10, 120][[1]] (* Vincenzo Librandi, Aug 29 2013 *)

log(18)/log(3) \\ Charles R Greathouse IV, Aug 06 2020

Equals 2+A102525. - R. J. Mathar, Oct 02 2023

A102447 Decimal expansion of log_3(20).

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Feb 23 2005

Hausdorff dimension of Menger sponge.

			2.72683302786084204139609463636416210490710364692981054479420028247...

Manfred Schroeder, Fractals, Chaos, Power Laws, Freeman,1991, p. 179.
Ian Stewart, Does God Play Dice?, The New Mathematics of Chaos, 2nd Ed., Blackwell Pub'l., Malden MA, 2002, p. 207.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
C. C. Bergemann, PlanetMath.org, Menger sponge
O. Knill, Menger Sponge
School of Mathematics and Statistics, University of St Andrews, Scotland, Abram Samoilovitch Besicovitch.
Turnbull WWW Server, Felix Hausdorff.
Eric Weisstein's World of Mathematics, Menger Sponge
Wikipedia, Hausdorff dimension.
Index entries for transcendental numbers

Cf. A100831, A102525, A113210, A152564.

Mathematica
```
RealDigits[ Log[3, 20], 10, 111][[1]]
```

log(20)/log(3) \\ Michel Marcus, Jul 19 2020

A153459 Decimal expansion of log_3 (6).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Oct 30 2009

Equals the Hausdorff dimension of Pascal's triangle modulo 3 (A083093). In general, the dimension of Pascal's triangle modulo a prime p is log(p*(p+1)/2) / log(p) (see Reiter link, theorem 2 page 117). - Bernard Schott, Dec 01 2022

			1.6309297535714574370995271143427608542995856401318804278706...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000.
A. M. Reiter, Determining the dimension of fractals generated by Pascal's triangle, Fibonacci Quarterly, 31(2), 1993, pp. 112-120.
Wikipedia, List of fractals by Hausdorff dimension (see Pascal triangle modulo 3).
Index entries for transcendental numbers

cf. A002391, A016629, A083093, A102525.

evalf(log(6)/log(3),80); # Bernard Schott, Dec 01 2022

RealDigits[Log[3, 6], 10, 120][[1]] (* Vincenzo Librandi, Aug 29 2013 *)

Equals A016629 / A002391 = 1 + A102525. - Bernard Schott, Dec 01 2022

A154196 Decimal expansion of log_3 (12).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Oct 30 2009

			2.2618595071429148741990542286855217085991712802637608557413...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for transcendental numbers

Cf. A102525, A153015.

RealDigits[Log[3, 12], 10, 120][[1]] (* Vincenzo Librandi, Aug 29 2013 *)

From Amiram Eldar, May 04 2023: (Start)
Equals 1 + 2 * log_3(2) = 1 + 2 * A102525.
Equasl 1/A153015. (End)

A346640 Decimal expansion of 2 - log_3(2).

Original entry on oeis.org

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

Offset: 1

Kevin Ryde, Jul 26 2021

Bedford and McMullen show this is the metric dimension of Hironaka's curve and equivalent carpets (see A346639).

			1.3690702464285425629004728856572391...

Timothy Bedford, Crinkly Curves, Markov Partitions and Dimension, Ph.D. thesis, University of Warwick, 1984, see proposition 4.1, page 89, cap(E) for the case s=2, t=2, r=3, Sum(k_i)=3 (and noting log(t) is a multiplier, not an exponent).
Curtis T. McMullen, Hausdorff Dimension of General Sierpinski Carpets, Nagoya Mathematical Journal, volume 96, number 19, 1984, pages 1-9, see page 2, m.dim(R) for the case m = s = 2 and n = r = 3.

Cf. A102525, A346639.

Cf. A213201.

RealDigits[2 - Log[3, 2], 10, 105][[1]] (* Amiram Eldar, Jul 27 2021 *)

2 - log(2)/log(3) \\ Michel Marcus, Jul 27 2021

Equals 2 - A102525.

Equals Sum_{d=1..2} d*log(1+1/d)/log(3). Compare with A213201. - Michel Marcus, Dec 25 2022

A228375 Decimal expansion of log_3(25).

Original entry on oeis.org

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

Offset: 1

Vincenzo Librandi, Aug 29 2013

			2.92994704143585433439408081535728079261586473333209937810578078959098...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for transcendental numbers

Cf. decimal expansion of log_3(m): A102525 (m=2), A100831 (m=4), A113209 (m=5), A153459 (m=6), A152565 (m=7), A113210 (m=8), A152566 (m=10), A154175 (m=11), A154196 (m=12), A154217 (m=13), A154463 (m=14), A154542 (m=15), A154751 (m=16), A154848 (m=17), A152549 (m=18), A155003 (m=19), A102447 (m=20), A155541 (m=21), A155694 (m=22), A155808 (m=23), A155922 (m=24), this sequence, A152564 (m=26).

Mathematica
```
RealDigits[Log[3, 25], 10, 100][[1]]
```

Equals 2*A113209. - R. J. Mathar, Sep 08 2013

cp's OEIS Frontend

A355512 Sum of numerator and denominator in the convergents of the approximation of log(2)/log(3) by a continued fraction.

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A355513 Sum of numerator and denominator in a rational approximation j/k of q = log(2)/log(3), such that abs(j/k - q) is a new minimum.

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A355515 Sum of numerator and denominator in a rational approximation j/k of q = log(2)/log(3), such that j/k - q is a new minimum, i.e., q is approximated from above.

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A152549 Decimal expansion of log_3(18).

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A102447 Decimal expansion of log_3(20).

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A153459 Decimal expansion of log_3 (6).

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A154196 Decimal expansion of log_3 (12).

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A346640 Decimal expansion of 2 - log_3(2).

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