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A007525 Decimal expansion of log_2 e.
(Formerly M3221)
6
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 (list; constant; graph; refs; listen; history; text; internal format)
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).
LINKS
Srinivasa Ramanujan, Question 769, J. Ind. Math. Soc.
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
Sequence in context: A064860 A091223 A242053 * A151966 A010778 A202322
KEYWORD
nonn,cons,easy
AUTHOR
STATUS
approved

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Last modified March 19 01:57 EDT 2024. Contains 370952 sequences. (Running on oeis4.)