A007525 Decimal expansion of log_2 e. (Formerly M3221)

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

1,2

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

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.

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)

Example

1.442695040888963407359924681...

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A064860 A091223 A242053 * A151966 A010778 A202322

Adjacent sequences: A007522 A007523 A007524 * A007526 A007527 A007528

Keyword

nonn,cons,easy

Author

N. J. A. Sloane

Status

approved