

A030178


Decimal expansion of LambertW(1): the solution to x*exp(x) = 1.


32



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
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OFFSET

0,1


COMMENTS

Sometimes called the Omega constant.
The first 59 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.  JeanFrançois Alcover, Dec 18 2013
Also the solution to x = log(x).  Robert G. Wilson v, Feb 22 2014


LINKS

Stanislav Sykora, Table of n, a(n) for n = 0..1999
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)
Eric Weisstein's World of Mathematics, Omega Constant
Eric Weisstein's World of Mathematics, Lambert WFunction


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 A267017 A021642
Adjacent sequences: A030175 A030176 A030177 * A030179 A030180 A030181


KEYWORD

nonn,cons,changed


AUTHOR

N. J. A. Sloane


STATUS

approved



