A202355 -id:A202355 - cp's OEIS frontend

A202357 Decimal expansion of the number x satisfying e*x = e^(-x).

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

Clark Kimberling, Dec 18 2011

See A202322 for a guide to related sequences. The Mathematica program includes a graph.

			0.2784645427610737951093587390229801554394774886...

Heine Halberstam and Hans Egon-Richert, Sieve Methods, Dover Publications (2011). See Theorem 2.1.

G. C. Greubel, Table of n, a(n) for n = 0..5000
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.
Index entries for transcendental numbers.

Cf. A202322, A215077, A218300.

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 *)

lambertw(exp(-1)) \\ Michel Marcus, Mar 21 2016

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

A202322 Decimal expansion of x satisfying x+2=exp(-x).

Original entry on oeis.org

4, 4, 2, 8, 5, 4, 4, 0, 1, 0, 0, 2, 3, 8, 8, 5, 8, 3, 1, 4, 1, 3, 2, 7, 9, 9, 9, 9, 9, 9, 3, 3, 6, 8, 1, 9, 7, 1, 6, 2, 6, 2, 1, 2, 9, 3, 7, 3, 4, 7, 9, 6, 8, 4, 7, 1, 7, 7, 3, 3, 0, 7, 6, 9, 8, 2, 0, 1, 5, 9, 9, 2, 1, 4, 2, 0, 0, 4, 0, 7, 8, 4, 9, 0, 8, 6, 5, 9, 2, 4, 8, 1, 7, 8, 7, 3, 9, 5, 5

Offset: 0

Clark Kimberling, Dec 18 2011

For many choices of u and v, there is just one value of x satisfying u*x+v=e^(-x).  Guide to related sequences, with graphs included in Mathematica programs:
u.... v.... x
1.... 2.... A202322
1.... 3.... A202323
2.... 2.... A202353
2.... e.... A202354
1... -1.... A202355
1.... 0.... A030178
2.... 0.... A202356
e.... 0.... A202357
3.... 0.... A202392
Suppose that f(x,u,v) is a function of three real variables and that g(u,v) is a function defined implicitly by f(g(u,v),u,v)=0.  We call the graph of z=g(u,v) an implicit surface of f.
For an example related to A202322, take f(x,u,v)=x+2-e^(-x) and g(u,v) = a nonzero solution x of f(x,u,v)=0.  If there is more than one nonzero solution, care must be taken to ensure that the resulting function g(u,v) is single-valued and continuous.  A portion of an implicit surface is plotted by Program 2 in the Mathematica section.

			x=-0.442854401002388583141327999999336819716262...

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for transcendental numbers

Cf. A202320.

(* Program 1:  A202322 *)
u = 1; v = 2;
f[x_] := u*x + v; g[x_] := E^-x
Plot[{f[x], g[x]}, {x, -1, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, -.45, -.44}, WorkingPrecision -> 110]
RealDigits[r]  (* A202322 *)
(* Program 2: implicit surface of u*x+v=e^(-x) *)
f[{x_, u_, v_}] := u*x + v - E^-x;
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 1, 2}]}, {v, 1, 3}, {u, 1, 3}];
ListPlot3D[Flatten[t, 1]] (* for A202322 *)
RealDigits[ ProductLog[E^2] - 2, 10, 99] // First (* Jean-François Alcover, Feb 14 2013 *)

lambertw(exp(2)) - 2 \\ G. C. Greubel, Jun 10 2017

x(u,v) = W(e^(v/u)/u) - v/u, where W = ProductLog = LambertW. - Jean-François Alcover, Feb 14 2013

Equals A226571 - 2 = LambertW(exp(2))-2. - Vaclav Kotesovec, Jan 09 2014

Digits from a(84) on corrected by Jean-François Alcover, Feb 14 2013

cp's OEIS Frontend

A202357 Decimal expansion of the number x satisfying e*x = e^(-x).

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