A007525 Decimal expansion of log_2 e.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• cons
• easy

Comments

Around 1670, James Gregory discovered by inversion of 1 - 1/2 + 1/3 - 1/4 + 1/5 - ... = log(2) that 1 + 1/2 - 1/12 + 1/24 - 19/720 + (27/1440 = 3/160) - 863/60480 + ... = 1/log(2). See formula with A002206 and A002207. See also A141417 signed /A091137; case i = 0 in A165313. First row in array p. 36 of the reference. - Paul Curtz, Sep 12 2011

This constant 1/log(2) is also related to the asymptotic evaluation of the maximum number of subtraction steps required to compute gcd(m, n) by the binary Euclidean algorithm, m and n being odd and chosen at random. - Jean-François Alcover, Jun 23 2014, after Steven Finch

Examples

			1.442695040888963407359924681...
		

References

• Paul Curtz, Intégration numérique des systèmes différentiels .. , note n° 12, Centre de Calcul Scientifique de l'Armement, Arcueil, 1969.
• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.18 Porter-Hensley constants, p. 159.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 25, equation 25:14:3 at page 232.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..5000
• Simon Plouffe, 1/log(2) the inverse of the natural logarithm of 2.
• Srinivasa Ramanujan, Question 769, J. Ind. Math. Soc.
• Index entries for transcendental numbers.

Crossrefs

Cf. A000670, A002206, A002207, A052882, A091137, A141417, A165313.

Programs

Mathematica

RealDigits[N[1/Log[2], 105]][[1]] (* Jean-François Alcover, Oct 30 2012 *)

PARI

1/log(2) \\ Charles R Greathouse IV, Jan 04 2016

Formula

Equals lim_{n->infinity} A000670(n)/A052882(n). - Mats Granvik, Aug 10 2009

Equals Sum_{k>=-1} A002206(k)/A002207(k). - Paul Curtz, Sep 12 2011

Also equals integral_{x>=2} 1/(x*log(x)^2). - Jean-François Alcover, May 24 2013

1/log(2) = Sum_{n = -infinity..infinity} (2^n / (1 + 2^2^n)). - Nicolas Nagel, Mar 16 2018

More generally: 1/log(2) = Sum_{n = -infinity..infinity} (2^(n+x) / (1 + 2^2^(n+x))) for all real x. - Nicolas Nagel, Jul 02 2019

From Amiram Eldar, Jun 04 2023: (Start)

Equals 1 + Sum_{k>=1} 1/(2^k * (1 + 2^(1/2^k))).

Equals Product_{k>=1} ((1 + 2^(1/2^k))/2). (End)