A020861 -id:A020861 - 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

A020862 Decimal expansion of log_2(10).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 55.

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

Cf. decimal expansion of log_2(m): A020857 (m=3), A020858 (m=5), A020859 (m=6), A020860 (m=7), A020861 (m=9), this sequence, 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).

RealDigits[Log[2, 10], 10, 100][[1]] (* Vincenzo Librandi, Aug 29 2013 *)

log(10)/log(2) \\ Charles R Greathouse IV, Aug 06 2020

Equals 1+A020858. - R. J. Mathar, Oct 25 2008

Definition improved by J. Lowell, May 03 2014

A020858 Decimal expansion of log_2(5).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Equals the Hausdorff dimension of the Sierpinski fractal square-based pyramid, when each square-based pyramid is replaced by 5 half-size such square-based pyramids (see IREM link). - Bernard Schott, Nov 30 2022

			2.3219280...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
IREM Paris-Nord, La pyramide de Sierpinski (in French).
Wikipedia, List of fractals by Hausdorff dimension (see Fractal pyramid).
Index entries for transcendental numbers

Cf. decimal expansion of log_2(m): A020857 (m=3), this sequence, 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).

Sierpinski pyramid: A000351 (number of pyramids), A279511 (number of vertices).

RealDigits[Log[2,5],10,120][[1]] (* Harvey P. Dale, Oct 20 2011 *)

log(5)/log(2) \\ Charles R Greathouse IV, Aug 06 2020

Definition improved by J. Lowell, May 03 2014

A155921 Decimal expansion of log_2(24) = 3+log_2(3).

Original entry on oeis.org

4, 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, Oct 30 2009

This is the third term in the sequence of real numbers discussed in A229168-A229170. - N. J. A. Sloane, Sep 28 2013

			4.5849625007211561814537389439478165087598144076924810604557...

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

Cf. decimal expansion of log_2(m): A020857 (m=3), 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), this sequence.

Cf. A229177.

RealDigits[Log[2,24],10,120][[1]] (* Harvey P. Dale, Dec 07 2011 *)

Equals 1 + A020864 = 2 + A020859 = 3 + A020857. - Jianing Song, Nov 16 2024

A154847 Decimal expansion of log_2 (17).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Oct 30 2009

			4.0874628412503394082540660108104043540112672823448206881266...

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

Cf. decimal expansion of log_2(m): A020857 (m=3), 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), this sequence, A154905 (m=18), A154995 (m=19), A155172 (m=20), A155536 (m=21), A155693 (m=22), A155793 (m=23), A155921 (m=24).

RealDigits[Log[2, 17], 10, 100][[1]] (* Vincenzo Librandi, Aug 29 2013 *)

A155172 Decimal expansion of log_2 (20).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Oct 30 2009

Equals 2 + A020858 = 1 + A020862 = A016643 / A002162. - Michel Marcus, Jul 28 2013

			4.3219280948873623478703194294893901758648313930245806120547...

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

Cf. decimal expansion of log_2(m): A020857 (m=3), 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), this sequence, A155536 (m=21), A155693 (m=22), A155793 (m=23), A155921 (m=24).

RealDigits[Log[2, 20], 10, 100][[1]] (* Vincenzo Librandi, Aug 29 2013 *)

A020863 Decimal expansion of log_2(11).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

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

Cf. decimal expansion of log_2(m): A020857 (m=3), A020858 (m=5), A020859 (m=6), A020860 (m=7), A020861 (m=9), A020862 (m=10), this sequence, 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).

RealDigits[Log[2,11],10,120][[1]] (* Harvey P. Dale, Nov 05 2011 *)

log(11)/log(2) \\ Charles R Greathouse IV, Aug 06 2020

Definition improved by J. Lowell, May 03 2014

A020864 Decimal expansion of log(12)/log(2).

Original entry on oeis.org

3, 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

			3.58496250072115618145373894394781650875981440.....

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

Cf. decimal expansion of log_2(m): A020857 (m=3), A020858 (m=5), A020859 (m=6), A020860 (m=7), A020861 (m=9), A020862 (m=10), A020863 (m=11), this sequence, 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).

RealDigits[Log[2, 12], 10, 100][[1]] (* Vincenzo Librandi, Aug 29 2013 *)

log(12)/log(2) \\ Charles R Greathouse IV, May 15 2019

Equals 1 + A020859. - R. J. Mathar, Oct 25 2008

Equals 2 + A020857 = -1 + A155921. - Jianing Song, Nov 16 2024

A152590 Decimal expansion of log_2(13).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Oct 28 2009

			3.7004397181410921603968126542566947336284364017910373695384...

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

Cf. decimal expansion of log_2(m): A020857 (m=3), A020858 (m=5), A020859 (m=6), A020860 (m=7), A020861 (m=9), A020862 (m=10), A020863 (m=11), A020864 (m=12), this sequence, 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).

RealDigits[Log[2, 13], 10, 100][[1]] (* Vincenzo Librandi, Aug 29 2013 *)

log(13)/log(2) \\ Charles R Greathouse IV, Aug 06 2020

A154462 Decimal expansion of log_2 (14).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Oct 30 2009

			3.8073549220576041074419693172318308086410266259661407836772...

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

Cf. decimal expansion of log_2(m): A020857 (m=3), A020858 (m=5), A020859 (m=6), A020860 (m=7), A020861 (m=9), A020862 (m=10), A020863 (m=11), A020864 (m=12), A152590 (m=13), this sequence, 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).

RealDigits[Log[2,14],10,120][[1]] (* Harvey P. Dale, Jul 19 2013 *)

Equals 1+A020860. - R. J. Mathar, Feb 01 2023

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A020857 Decimal expansion of log_2(3).

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A020862 Decimal expansion of log_2(10).

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A020858 Decimal expansion of log_2(5).

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A155921 Decimal expansion of log_2(24) = 3+log_2(3).

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A154847 Decimal expansion of log_2 (17).

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A155172 Decimal expansion of log_2 (20).

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A020863 Decimal expansion of log_2(11).

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