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

1,2

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

References

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.16, p. 145.

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.

Sequence in context: A021635 A021175 A011095 * A096413 A334116 A222591

Adjacent sequences: A020854 A020855 A020856 * A020858 A020859 A020860

Keyword

nonn,cons,changed

Author

N. J. A. Sloane

Extensions

Comment generalized by J. Lowell, Apr 26 2014

Status

approved