A102525 -id:A102525 - cp's OEIS frontend

A020857 Decimal expansion of log_2(3).

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

Offset: 1

N. J. A. Sloane

The fractional part of the binary logarithm of 3 * 2^n (A007283) is the same as that of any number of the form log_2 (A007283(n)) (e.g., log_2(192) = 7.5849625...). Furthermore, a necessary but not sufficient condition for a number to be Fibbinary (A003714) is that the fractional part of its binary logarithm does not exceed that of this number. - Alonso del Arte, Jun 22 2012
Log_2(3)-1 = 0.58496... is the exponent in n^(log_2(3)-1), the asymptotic growth rate of the number of odd coefficients in (1+x)^n mod 2 (Cf. Steven Finch ref.). - Jean-François Alcover, Aug 13 2014
Equals the Hausdorff dimension of the Sierpiński triangle. - Stanislav Sykora, May 27 2015
The complexity of Karatsuba algorithm for the multiplication of two n-digit numbers is O(n^log_2(3)). - Jianing Song, Apr 28 2019

			log_2(3) = 1.5849625007211561814537389439...

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 24, 257.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.16, p. 145.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
E. G. Dunne, Pianos and Continued Fractions
Shalom Eliahou, Le problème 3n+1 : y a-t-il des cycles non triviaux? (III), Images des Mathématiques, CNRS, 2011 (in French).
Steven Finch, Pascal Sebah and Zai-Qiao Bai, Odd Entries in Pascal's Trinomial Triangle, arXiv:0802.2654 [math.NT], 2008, p. 1.
Karatsuba, The Complexity of Computations, Proceedings of the Steklov Institute of Mathematics, 1995: 169-183.
Youngik Lee, Numerical Approach on Collatz Conjecture, Preprints.org, Brown Univ., 2024. See p. 13.
Simon Plouffe, log(3)/log(2) to 10000 digits
A. M. Reiter, Determining the dimension of fractals generated by Pascal's triangle, Fibonacci Quart, 31(2):112-120, 1993.
Eric Weisstein's World of Mathematics, Stolarsky-Harborth Constant
Eric Weisstein's World of Mathematics, Pascal's Triangle
Eric Weisstein's World of Mathematics, Sierpiński Sieve
Wikipedia, Karatsuba algorithm
Wikipedia, Sierpinski triangle
Index entries for transcendental numbers

Cf. decimal expansion of log_2(m): this sequence, A020858 (m=5), A020859 (m=6), A020860 (m=7), A020861 (m=9), A020862 (m=10), A020863 (m=11), A020864 (m=12), A152590 (m=13), A154462 (m=14), A154540 (m=15), A154847 (m=17), A154905 (m=18), A154995 (m=19), A155172 (m=20), A155536 (m=21), A155693 (m=22), A155793 (m=23), A155921 (m=24).

Cf. A102525.

evalf(log[2](3), 100); # Bernard Schott, Feb 02 2023

RealDigits[Log[2, 3], 10, 100][[1]] (* Alonso del Arte, Jun 22 2012 *)

log(3)/log(2) \\ Michel Marcus, Jan 11 2016

Equals 1 / A102525. - Bernard Schott, Feb 02 2023

Comment generalized by J. Lowell, Apr 26 2014

A154009 Decimal expansion of log_6 (9).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Oct 30 2009

			1.2262943855309168262595077230643582470697162810857931432210...

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

Cf. decimal expansion of log_6(m): A152683 (m=2), A152935 (m=3), A153102 (m=4), A153202 (m=5), A153617 (m=7), A153754 (m=8), this sequence, A154157 (m=10), A154178 (m=11), A154199 (m=12), A154278 (m=13), A154466 (m=14), A154567 (m=15), A154776 (m=16), A154856 (m=17), A154911 (m=18), A155044 (m=19), A155490 (m=20), A155554 (m=21), A155697 (m=22), A155823 (m=23), A155959 (m=24).

SetDefaultRealField(RealField(100)); Log(9)/Log(6); // G. C. Greubel, Sep 14 2018

RealDigits[Log[6, 9], 10, 100][[1]] (* Vincenzo Librandi, Aug 31 2013 *)

default(realprecision, 100); log(9)/log(6) \\ G. C. Greubel, Sep 14 2018

Equals A016632 / A016629 =2/(1+A102525). - R. J. Mathar, Jul 29 2024

A020915 Number of digits in base-3 representation of 2^n.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 7, 8, 9, 9, 10, 11, 11, 12, 12, 13, 14, 14, 15, 16, 16, 17, 18, 18, 19, 19, 20, 21, 21, 22, 23, 23, 24, 24, 25, 26, 26, 27, 28, 28, 29, 30, 30, 31, 31, 32, 33, 33, 34, 35, 35, 36, 36, 37, 38, 38, 39, 40, 40, 41, 42, 42, 43, 43, 44, 45, 45, 46, 47

Offset: 0

Clark Kimberling

For n > 0, first differences of A022331. - Michel Marcus, Oct 03 2013

William A. Tedeschi, Table of n, a(n) for n = 0..10000
Mike Winkler, The algorithmic structure of the finite stopping time behavior of the 3x+1 function, arXiv:1709.03385 [math.GM], 2017.

Cf. A007089, A020914, A076227, A081604, A000079.
Cf. A022331, A102525, A136409.
Cf. A022924 (first differences).

a020915 = a081604 . a000079  -- Reinhard Zumkeller, Mar 28 2015

[Round(1+Floor(n*(Log(2))/Log(3))): n in [0..80]]; // Vincenzo Librandi, Dec 05 2015

Table[Length[IntegerDigits[2^n,3]],{n,0,80}] (* Harvey P. Dale, May 02 2012 *)
Table[1 + Floor[n*Log[3, 2]], {n, 0, 73}] (* L. Edson Jeffery, Dec 04 2015 *)
IntegerLength[2^Range[0,80],3] (* Harvey P. Dale, Nov 17 2022 *)

a(n)=logint(2^n,3)+1 \\ Charles R Greathouse IV, Sep 02 2015

a(n) = 1 + floor(n*log_3(2)) = 1 + floor(n*A102525) = 1 + A136409(n). - R. J. Mathar, May 23 2009
a(n) = A081604(A000079(n)). - Reinhard Zumkeller, Jul 12 2011
a(A020914(n)) = n + 1. - Reinhard Zumkeller, Mar 28 2015

More terms from James Sellers

A117630 Complement of A056576.

Original entry on oeis.org

2, 5, 8, 10, 13, 16, 18, 21, 24, 27, 29, 32, 35, 37, 40, 43, 46, 48, 51, 54, 56, 59, 62, 65, 67, 70, 73, 75, 78, 81, 83, 86, 89, 92, 94, 97, 100, 102, 105, 108, 111, 113, 116, 119, 121, 124, 127, 130, 132, 135, 138, 140, 143, 146, 149, 151, 154, 157, 159, 162, 165

Offset: 1

Robert G. Wilson v, Apr 08 2006

nonn

A Beatty sequence.

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for sequences related to Beatty sequences

Cf. A054414, A056576.
Cf. A102525 (decimal expansion of log_3(2)).
Cf. A254312 (sequence arises as exponents in array definition).

[Floor(n*Log(3)/Log(3/2)): n in [1..80]]; // Vincenzo Librandi, Apr 17 2015

seq(floor(n*log[3/2](3)), n=1..100); # Robert Israel, Nov 09 2015

Mathematica
```
Table[Floor[n*Log[3/2, 3]], {n, 61}]
```

vector(100, n, floor(n*log(3)/log(3/2))) \\ Altug Alkan, Nov 10 2015

from operator import sub
from sympy import integer_log
def A117630(n):
    def f(x): return n+sub(*integer_log(1<Chai Wah Wu, Oct 09 2024

a(n) = floor(n*log(3)/log(3/2)).

a(n) = A054414(n) - 1. - Ruud H.G. van Tol, May 10 2024

A136409 a(n) = floor(n*log_3(2)).

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 3, 4, 5, 5, 6, 6, 7, 8, 8, 9, 10, 10, 11, 11, 12, 13, 13, 14, 15, 15, 16, 17, 17, 18, 18, 19, 20, 20, 21, 22, 22, 23, 23, 24, 25, 25, 26, 27, 27, 28, 29, 29, 30, 30, 31, 32, 32, 33, 34, 34, 35, 35, 36, 37, 37, 38, 39, 39, 40, 41, 41, 42, 42, 43, 44, 44, 45, 46, 46

Offset: 0

Ctibor O. Zizka, Mar 31 2008

a(n) is the exponent of the greatest power of 3 not exceeding 2^n.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Titu Andreescu and Dorin Andrica, On a class of sums involving the floor function, Mathematical Reflections 3, 2006.
H. W. Gould and Jocelyn Quaintance, Floor and Roof Function Analogs of the Bell Numbers, INTEGERS: El. J. Comb. Number Theory 7 (2007), #A58.

Cf. A102525, A156301.

a136409 = floor . (* logBase 3 2) . fromIntegral
-- Reinhard Zumkeller, Jul 03 2015

With[{k = Log[3, 2]}, Array[Floor[k #] &, 75, 0]] (* Michael De Vlieger, Sep 29 2019 *)

a(n)=logint(2^n,3) \\ Charles R Greathouse IV, Sep 02 2015

from sympy import integer_log
def A136409(n): return integer_log(1<Chai Wah Wu, Oct 09 2024

From Benjamin Lombardo, Sep 08 2019: (Start)
a(A020914(k)) = k.
a(A054414(k)) = a(A054414(k)-1) for k > 0. (End)

A121384 a(n) = ceiling(n*e).

Original entry on oeis.org

0, 3, 6, 9, 11, 14, 17, 20, 22, 25, 28, 30, 33, 36, 39, 41, 44, 47, 49, 52, 55, 58, 60, 63, 66, 68, 71, 74, 77, 79, 82, 85, 87, 90, 93, 96, 98, 101, 104, 107, 109, 112, 115, 117, 120, 123, 126, 128, 131, 134, 136, 139, 142, 145, 147, 150, 153, 155, 158, 161, 164, 166

Offset: 0

Mohammad K. Azarian, Sep 06 2006

Because the difference between e=A001113 and the constant 1/(1-theta), theta = A102525, defined in A054414 is only 0.00877, the difference |a(n)-A054414(n)| increases approximately as 0.00877*n. - R. J. Mathar, Apr 14 2008

A022843(n) <= A022852(n) <= a(n). - Reinhard Zumkeller, Mar 17 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A022843, A022852.

a121384 = ceiling . (* exp 1) . fromIntegral
-- Reinhard Zumkeller, Mar 17 2015

Table[Ceiling[n * E], {n, 0, 80}] (* Vincenzo Librandi, Feb 22 2013 *)

A355514 Sum of numerator and denominator in a rational approximation j/k of q = log(2)/log(3), such that q - j/k is a new minimum, i.e., q is approximated from below.

Original entry on oeis.org

1, 3, 8, 13, 44, 75, 106, 243, 380, 517, 654, 791, 2510, 4229, 5948, 7667, 9386, 11105, 12824, 14543, 16262, 17981, 19700, 21419, 23138, 24857, 26576, 28295, 30014, 31733, 33452, 35171, 36890, 38609, 40328, 122703, 205078, 492531, 27869189, 166722603, 305576017

Offset: 1

Hugo Pfoertner, Jul 05 2022

nonn

Cf. A102525, A355512, A355513, A355515.

Terms are candidates for being in A355240, which shares 3, 8, 13, 44, 75.

a355514(upto) = {my(q=log(2)/log(3), dmin=oo);for (m=1, upto, my(n=floor(m*q), qq=n/m, d=q-qq); if (d

A100831 Decimal expansion of log(4)/log(3).

Original entry on oeis.org

1, 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, 1, 1, 2, 2, 7, 0

Offset: 1

Lekraj Beedassy, Jan 07 2005

log_3(4) is the Hausdorff dimension of the Koch snowflake.

A transcendental number. Also the Hausdorff dimension of 2D Cantor dust (for N-dimensional Cantor dust, see A102525). - Stanislav Sykora, Apr 19 2016

			log(4)/log(3) = 1.26185950714291487419905422868552170859917128...

Martin Gardner, Aha! Gotcha!, "A Pathological Curve", W. H. Freeman, NY, 1982, p. 77.
Martin Gardner's Sixth Book of Mathematical Diversions from Scientific American, University of Chicago Press, IL, 1983, p. 227.
Martin Gardner, The Colossal Book of Mathematics, W. W. Norton, NY, 2001, p. 322.
Nigel Lesmoir-Gordon, Will Rood and Ralph Edney, Introducing Fractal Geometry, Totem Books USA, Lanham, MD, 2001, p. 28.
Manfred Schroeder, Fractals, Chaos, Power Laws, Freeman, 1991, p. 177.
David Wells, The Penguin Dictionary of Curious and Interesting Geometry, Penguin Books, 1991, pp. 135-136.

V. L. Almstrum, Visual Koch (Applet).
Robert Ferreol and Jacques Mandonnet, Koch's Curve.
Florida Atlantic University, Koch's Curve Applet.
P. Kernan, Koch Snowflake.
Kris, Koch Fractal,Koch Snowflake.
Aaron Krowne, PlanetMath.org, Koch curve.
M. L. Lapidus & E. P. J. Pearse, A tube formula for the Koch snowflake curve,with applications to complex dimensions, arXiv:math-ph/0412029, 2004-2005.
Simon Plouffe, log4/log3 to 10000 digits.
Larry Riddle, Koch Curve.
Alain Schuler, Chaos and fractal:the Koch's curve.
Gerard Villemin, Almanac of Numbers, Koch's Curve or Snowflake.
Eric Weisstein's World of Mathematics, Koch Snowflake.
Eric Weisstein's World of Mathematics, Cantor Dust.
Wikipedia, Koch curve.
Index entries for transcendental numbers

Cf. A094148, A102525.

RealDigits[Log[3, 4], 10, 111][[1]] (* Robert G. Wilson v, Jan 07 2005 *)

log(4)/log(3) \\ Altug Alkan, Apr 19 2016

Equals 2*A102525. - Stanislav Sykora, Apr 19 2016

More terms from Robert G. Wilson v, Jan 07 2005

A152566 Decimal expansion of log_3(10).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Oct 28 2009

			2.0959032742893846042965675220214012506075180067979301169235...

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

Cf. A102524, A100831, A113209, A102525, A152565, A113210, A152566, A152549, A152564.

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

log(10)/log(3) \\ Charles R Greathouse IV, Aug 06 2020

A156301 a(n) = ceiling( n * log_3(2) ) = ceiling(n * 0.6309297535714574371...).

Original entry on oeis.org

0, 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 7, 8, 9, 9, 10, 11, 11, 12, 12, 13, 14, 14, 15, 16, 16, 17, 18, 18, 19, 19, 20, 21, 21, 22, 23, 23, 24, 24, 25, 26, 26, 27, 28, 28, 29, 30, 30, 31, 31, 32, 33, 33, 34, 35, 35, 36, 36, 37, 38, 38, 39, 40, 40, 41, 42, 42, 43, 43, 44, 45, 45, 46, 47

Offset: 0

Jonathan Vos Post, Feb 07 2009

a(n) is the unique k such that 1/2 <= 3^a(n)/2^(n+1) < 3/2. Equality occurs iff n = 0. See Gorman-Huang and Marks.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Li an-Ping, A note on the counterfeit coins problem, arXiv:0902.0841 [math.CO], 2009-2010.
A. Gorman-Huang and E. Marks, Approximating Powers of 2 Using Powers of 3 and Break Point Rounding, Girls' Angle Bulletin, Vol. 18, No. 5 (2025), 8-10.

Cf. A020915, A102525. - R. J. Mathar, Feb 19 2009

Cf. A136409.

a156301 = ceiling . (* logBase 3 2) . fromIntegral
-- Reinhard Zumkeller, Jul 03 2015

seq(ceil(n*log[3](2)),n=0..120) ; # R. J. Mathar, Mar 14 2009

With[{c=Log[3,2]},Ceiling[c*Range[0,80]]] (* Harvey P. Dale, Aug 07 2015 *)

from operator import sub
from sympy import integer_log
def A156301(n): return sub(*integer_log(1<Chai Wah Wu, Oct 09 2024

More terms from R. J. Mathar, Mar 14 2009

Edited by N. J. A. Sloane, May 23 2009 at the suggestion of Hagen von Eitzen

cp's OEIS Frontend

A020857 Decimal expansion of log_2(3).

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A154009 Decimal expansion of log_6 (9).

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A020915 Number of digits in base-3 representation of 2^n.

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A117630 Complement of A056576.

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A136409 a(n) = floor(n*log_3(2)).

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A121384 a(n) = ceiling(n*e).

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A355514 Sum of numerator and denominator in a rational approximation j/k of q = log(2)/log(3), such that q - j/k is a new minimum, i.e., q is approximated from below.