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

Author

N. J. A. Sloane

Keywords

• nonn
• cons

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

Examples

			log_2(3) = 1.5849625007211561814537389439...
		

References

• 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.

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

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.

Programs

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 *)

PARI

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

Formula

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

Extensions

Comment generalized by J. Lowell, Apr 26 2014