A154847 -id:A154847

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A020857

Decimal expansion of log_2(3).

+10
48

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

LINKS

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

FORMULA

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

EXAMPLE

log_2(3) = 1.5849625007211561814537389439...

MAPLE

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

MATHEMATICA

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

PROG

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

CROSSREFS

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.

KEYWORD

nonn,cons

AUTHOR

N. J. A. Sloane

EXTENSIONS

Comment generalized by J. Lowell, Apr 26 2014

STATUS

approved

A020862

Decimal expansion of log_2(10).

+10
26

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

REFERENCES

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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Index entries for transcendental numbers.

FORMULA

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

MATHEMATICA

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

PROG

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

CROSSREFS

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).

KEYWORD

nonn,cons

AUTHOR

N. J. A. Sloane

EXTENSIONS

Definition improved by J. Lowell, May 03 2014

STATUS

approved

A020858

Decimal expansion of log_2(5).

+10
22

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

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

EXAMPLE

2.3219280...

MATHEMATICA

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

PROG

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

CROSSREFS

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).

KEYWORD

nonn,cons

AUTHOR

N. J. A. Sloane

EXTENSIONS

Definition improved by J. Lowell, May 03 2014

STATUS

approved

A155921

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

+10
21

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Index entries for transcendental numbers

EXAMPLE

4.5849625007211561814537389439478165087598144076924810604557...

MATHEMATICA

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

CROSSREFS

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.

KEYWORD

nonn,cons

AUTHOR

N. J. A. Sloane, Oct 30 2009

STATUS

approved

A154858

Decimal expansion of log_8 (17).

+10
19

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Index entries for transcendental numbers

FORMULA

Equals A016640 / A016631 = A154847/3. - R. J. Mathar, Apr 11 2024

EXAMPLE

1.3624876137501131360846886702701347846704224274482735627088...

MATHEMATICA

RealDigits[Log[8, 17], 10, 100][[1]] (* Vincenzo Librandi, Sep 01 2013 *)

PROG

(PARI) default(realprecision, 100); log(17)/log(8) \\ G. C. Greubel, Aug 31 2018

(Magma) SetDefaultRealField(RealField(100)); Log(17)/Log(8); // G. C. Greubel, Aug 31 2018

CROSSREFS

Cf. decimal expansion of log_8(m): A152956 (m=3), A153204 (m=5), A153493 (m=6), A153618 (m=7), A154010 (m=9), A154159 (m=10), A154180 (m=11), A154201 (m=12), A154309 (m=13), A154468 (m=14), A154574 (m=15), this sequence, A154927 (m=18), A155060 (m=19), A155502 (m=20), A155675 (m=21), A155741 (m=22), A155827 (m=23), A155975 (m=24).

KEYWORD

nonn,cons

AUTHOR

N. J. A. Sloane, Oct 30 2009

STATUS

approved

A155172

Decimal expansion of log_2 (20).

+10
18

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Index entries for transcendental numbers

EXAMPLE

4.3219280948873623478703194294893901758648313930245806120547...

MATHEMATICA

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

CROSSREFS

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).

KEYWORD

nonn,cons

AUTHOR

N. J. A. Sloane, Oct 30 2009

STATUS

approved

A020861

Decimal expansion of log_2(9)

+10
17

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Index entries for transcendental numbers.

FORMULA

Equals 2 * A020857. - Amiram Eldar, Jun 25 2023

EXAMPLE

3.16992500144231236290747788789563301751962881538496...

MATHEMATICA

RealDigits[Log[9]/Log[2], 10, 120][[1]] (* Harvey P. Dale, Jul 08 2013 *)

RealDigits[Log2[9], 10, 120][[1]] (* Harvey P. Dale, May 21 2023 *)

PROG

(PARI) log(9)/log(2) \\ Charles R Greathouse IV, Aug 06 2020

CROSSREFS

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

KEYWORD

nonn,cons

AUTHOR

N. J. A. Sloane

EXTENSIONS

Definition improved by J. Lowell, May 03 2014

STATUS

approved

A020863

Decimal expansion of log_2(11).

+10
17

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Index entries for transcendental numbers

MATHEMATICA

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

PROG

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

CROSSREFS

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).

KEYWORD

nonn,cons

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Definition improved by J. Lowell, May 03 2014

STATUS

approved

A020864

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

+10
17

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Index entries for transcendental numbers

FORMULA

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

EXAMPLE

3.58496250072115618145373894394781650875981440.....

MATHEMATICA

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

PROG

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

CROSSREFS

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).

KEYWORD

nonn,cons

AUTHOR

N. J. A. Sloane.

STATUS

approved

A152590

Decimal expansion of log_2(13).

+10
17

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Index entries for transcendental numbers

EXAMPLE

3.7004397181410921603968126542566947336284364017910373695384...

MATHEMATICA

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

PROG

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

CROSSREFS

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).

KEYWORD

nonn,cons

AUTHOR

N. J. A. Sloane, Oct 28 2009

STATUS

approved

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