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A030178 Decimal expansion of LambertW(1): the solution to x*exp(x) = 1. 36
5, 6, 7, 1, 4, 3, 2, 9, 0, 4, 0, 9, 7, 8, 3, 8, 7, 2, 9, 9, 9, 9, 6, 8, 6, 6, 2, 2, 1, 0, 3, 5, 5, 5, 4, 9, 7, 5, 3, 8, 1, 5, 7, 8, 7, 1, 8, 6, 5, 1, 2, 5, 0, 8, 1, 3, 5, 1, 3, 1, 0, 7, 9, 2, 2, 3, 0, 4, 5, 7, 9, 3, 0, 8, 6, 6, 8, 4, 5, 6, 6, 6, 9, 3, 2, 1, 9, 4, 4, 6, 9, 6, 1, 7, 5, 2, 2, 9, 4, 5, 5, 7, 6, 3, 8 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Sometimes called the Omega constant.

The first 58 digits form a prime: 5671432904097838729999686622103555497538157871865125081351. - Jonathan Vos Post, Nov 09 2004

LambertW(n)/n, n=1,2,3,4,5,..., can be calculated with the same recurrence as for the numerators in Dirichlet series for logarithms of n using tetration. Convergence is slow for large numbers. See Mathematica program for recurrence. Tetration appears to work also for LambertW(n*(complex number))/n, n=1,2,3,4,5,... See link to mathematics stackexchange. (Conjecture.) - Mats Granvik, Oct 19 2013

Infinite power tower for c = 1/E, i.e., c^c^c^..., where c = 1/A068985. - Stanislav Sykora, Nov 03 2013

Notice the narrow interval exp(-gamma) < w(1) < gamma, with gamma = A001620. - Jean-Fran├žois Alcover, Dec 18 2013

Also the solution to x = -log(x). - Robert G. Wilson v, Feb 22 2014

REFERENCES

Stanislav Sykora, Fixed points of the mappings exp(z) and -exp(z) in C, http://www.ebyte.it/library/docs/math16/2016_MATH_Sykora_FixedPointsExp.pdf; DOI: 10.3247/SL6Math16.002, 2016.

LINKS

G. C. Greubel and Stanislav Sykora, Table of n, a(n) for n = 0..10000 (terms 0..1999 from Stanislav Sykora)

blackpenredpen, Finding Omega, featuring Newton's method, video (2018)

Daniel Cummerow, Sound of Mathematics, Constants.

Mats Granvik, LambertW(k)/k by tetration for natural numbers

Simon Plouffe, Lambert W(1, 0)

Simon Plouffe, The omega constant or W(1)

Michael A. Sherbon, "Fine-Structure Constant from Golden Ratio Geometry", International Journal of Mathematics and Physical Sciences Research (2018) Vol. 5, Issue 2, pp. 89-100.

Eric Weisstein's World of Mathematics, Omega Constant

Eric Weisstein's World of Mathematics, Lambert W-Function

FORMULA

1/A030797.

EXAMPLE

0.5671432904097838729999686622103555497538157871865125081351310792230457930866...

MAPLE

evalf(LambertW(1));

MATHEMATICA

RealDigits[ ProductLog[1], 10, 111][[1]] (* Robert G. Wilson v, May 19 2004 *)

PROG

(PARI) solve(x=0, 1, x*exp(x)-1) \\ Charles R Greathouse IV, Mar 20 2012

(PARI) lambertw(1) \\ Stanislav Sykora, Nov 03 2013

CROSSREFS

Cf. A019474, A059526, A059527.

Cf. A276759 (another fixed point of -exp(z)).

Sequence in context: A081820 A214681 A019978 * A038458 A284361 A267017

Adjacent sequences:  A030175 A030176 A030177 * A030179 A030180 A030181

KEYWORD

nonn,cons

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified September 25 21:12 EDT 2018. Contains 315425 sequences. (Running on oeis4.)