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A007525 Decimal expansion of log_2 e.
(Formerly M3221)
4
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

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.

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

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

Adjacent sequences:  A007522 A007523 A007524 * A007526 A007527 A007528

KEYWORD

nonn,cons,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified December 16 04:19 EST 2018. Contains 318158 sequences. (Running on oeis4.)