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A202357
Decimal expansion of the number x satisfying e*x = e^(-x).
11
2, 7, 8, 4, 6, 4, 5, 4, 2, 7, 6, 1, 0, 7, 3, 7, 9, 5, 1, 0, 9, 3, 5, 8, 7, 3, 9, 0, 2, 2, 9, 8, 0, 1, 5, 5, 4, 3, 9, 4, 7, 7, 4, 8, 8, 6, 1, 9, 7, 4, 5, 7, 6, 5, 4, 5, 3, 1, 7, 8, 1, 0, 5, 5, 3, 5, 0, 2, 9, 3, 7, 5, 4, 5, 9, 9, 4, 9, 8, 9, 8, 1, 9, 2, 0, 4, 9, 8, 4, 2, 8, 1, 1, 2, 9, 9, 4, 2, 8
OFFSET
0,1
COMMENTS
See A202322 for a guide to related sequences. The Mathematica program includes a graph.
REFERENCES
Heine Halberstam and Hans Egon-Richert, Sieve Methods, Dover Publications (2011). See Theorem 2.1.
LINKS
W. Gautschi, The incomplete Gamma Function since Tricomi, Atti Conv. Lincei 147 (1999) 203, eq. (2.16).
H. J. H. Tuenter, On the generalized Poisson distribution, arXiv:math/0606238 [math.ST], 2006. Published version on On the Generalized Poisson Distribution, Statistica Neerlandica, 54(3):374-376, November 2000.
S. M. Zemyan, On the zeros of the Nth partial sum of the exponential series, Am. Math. Monthly 112 (2005) 891-909.
FORMULA
The constant in A202355 minus 1. - R. J. Mathar, Dec 21 2011
1+x+log(x)=0. - R. J. Mathar, Nov 02 2012
Equals LambertW(exp(-1)). - Vaclav Kotesovec, Jan 10 2014
EXAMPLE
x=0.2784645427610737951093587390229801554394774886...
MATHEMATICA
u = E; v = 0;
f[x_] := u*x + v; g[x_] := E^-x
Plot[{f[x], g[x]}, {x, 0, 1}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .27, .28}, WorkingPrecision -> 110]
RealDigits[r] (* A202357 *)
RealDigits[ ProductLog[1/E], 10, 99] // First (* Jean-François Alcover, Feb 14 2013 *)
RealDigits[LambertW[Exp[-1]], 10, 120][[1]] (* Harvey P. Dale, Dec 24 2019 *)
PROG
(PARI) lambertw(exp(-1)) \\ Michel Marcus, Mar 21 2016
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Dec 18 2011
STATUS
approved