A202322 -id:A202322 - cp's OEIS frontend

A202320 Decimal expansion of x < 0 satisfying x + 2 = e^x, negated.

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

Offset: 1

Clark Kimberling, Dec 16 2011

For many choices of u and v, there is just one x < 0 and one x > 0 satisfying u*x + v = exp(x). Guide to related sequences, with graphs included in Mathematica programs:
u.... v.... least x.....greatest x
1.... 2.... A202320.... A202321
1.... 3.... A202324.... A202325
2.... 1.... ..(x=0).... A202343
3.... 1.... ..(x=0).... A202344
2.... 2.... A202345.... A202346
1.... e.... A202347.... A104689
e.... 1.... ..(x=0).... A202350
3.... 0.... A202351.... A202352
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 A202320, take f(x,u,v) = u*x + v - exp(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.
The solution for u*x + v = exp(x) is -LambertW(-1/(u*exp(v/u))) - v/u. - Andrea Pinos, Sep 14 2023

			x < 0:  -1.841405660436960637846604658012486...
x > 0:   1.1461932206205825852370610285213682...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Wikipedia, Lambert W function. Applications
Index entries for transcendental numbers.

Cf. A202322, A202348.

(* Program 1:  A202320 and A202321 *)
u = 1; v = 2;
f[x_] := u*x + v; g[x_] := E^x
Plot[{f[x], g[x]}, {x, -2, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, -1.9, -1.8}, WorkingPrecision -> 110]
RealDigits[r]  (* A202320 *)
r = x /. FindRoot[f[x] == g[x], {x, 1.1, 1.2}, WorkingPrecision -> 110]
RealDigits[r]  (* A202321 *)
(* 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, 2, 4}, {u, 2, 4}];
ListPlot3D[Flatten[t, 1]] (* for A202320 *)
RealDigits[-ProductLog[-1/E^2] - 2, 10, 99] // First (* Jean-François Alcover, Feb 26 2013 *)

solve(x=-2, -1, x + 2 - exp(x)) \\ Michel Marcus, Dec 30 2018

lambertw(-exp(-2))-2 \\ Charles R Greathouse IV, Feb 03 2025

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

Equals 2 - A202348. - Jianing Song, Dec 30 2018

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

Original entry on oeis.org

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

A202356 Decimal expansion of the number x satisfying 2x=exp(-x).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Dec 18 2011

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

			x=0.35173371124919582602490930092995106517146...

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

Cf. A202322.

u = 2; v = 0;
f[x_] := u*x + v; g[x_] := E^-x
Plot[{f[x], g[x]}, {x, 0, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .35, .36}, WorkingPrecision -> 110]
RealDigits[r] (* A202356 *)
RealDigits[ ProductLog[1/2], 10, 99] // First (* Jean-François Alcover, Feb 14 2013 *)

lambertw(1/2) \\ G. C. Greubel, Jun 10 2017

A202355 Decimal expansion of the number x satisfying x-1=exp(-x).

Original entry on oeis.org

1, 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

Offset: 1

Clark Kimberling, Dec 18 2011

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

			x=1.2784645427610737951093587390229801554394...

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A202322.

u = 1; v = -1;
f[x_] := u*x + v; g[x_] := E^-x
Plot[{f[x], g[x]}, {x, 0, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, 1.2, 1.3}, WorkingPrecision -> 110]
RealDigits[r] (* A202355 *)
(* other program *)
RealDigits[1 + ProductLog[1/E], 10, 99] // First (* Jean-François Alcover, Feb 14 2013 *)

1 + lambertw(exp(-1)) \\ G. C. Greubel, Jun 10 2017

Equals A202357 + 1. - Vaclav Kotesovec, Jan 31 2015

A202323 Decimal expansion of the number x satisfying x+3=exp(-x).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Dec 18 2011

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

			x=-0.7920599684306770014183958778854220...

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

Cf. A202322.

u = 3; v = 0;
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, 0.6, 0.7}, WorkingPrecision -> 110]
RealDigits[r] (* A202351 *)
r = x /. FindRoot[f[x] == g[x], {x, 1.5, 1.6}, WorkingPrecision -> 110]
RealDigits[r] (* A202352 *)
(* other program *)
RealDigits[ ProductLog[E^3] - 3, 10, 99] // First (* Jean-François Alcover, Feb 14 2013 *)

lambertw(exp(3)) - 3 \\ G. C. Greubel, Jun 10 2017

a(97)-a(98) corrected by Jean-François Alcover, Feb 14 2013

A202353 Decimal expansion of the number x satisfying 2*x + 2 = exp(-x), negated.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Dec 18 2011

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

			x = -0.3149230578454060539717505194623698115859...

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

Cf. A202322.

u = 2; v = 2;
f[x_] := u*x + v; g[x_] := E^-x
Plot[{f[x], g[x]}, {x, -1, 1}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, -.4, -.3}, WorkingPrecision -> 110]
RealDigits[r] (* A202353 *)
(* other program *)
RealDigits[ ProductLog[E/2] - 1, 10, 99] // First (* Jean-François Alcover, Feb 14 2013 *)

lambertw(exp(1)/2) - 1 \\ G. C. Greubel, Jun 09 2017

Equals W(e/2) - 1, where W(x) is the Lambert W-function. - G. C. Greubel, Jun 09 2017

A202354 Decimal expansion of the number x satisfying x+e=exp(-x).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Dec 18 2011

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

			x=-0.7015020635668446110824969171586507639...

G. C. Greubel, Table of n, a(n) for n = 0..5000

Cf. A202322.

u = 1; v = E;
f[x_] := u*x + v; g[x_] := E^-x
Plot[{f[x], g[x]}, {x, -1, 1}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, -.8, -.7}, WorkingPrecision -> 110]
RealDigits[r] (* A202354 *)
(* other program *)
RealDigits[ ProductLog[E^E] - E , 10, 99] // First (* Jean-François Alcover, Feb 14 2013 *)

lambertw(exp(exp(1))) - exp(1) \\ G. C. Greubel, Jun 10 2017

A202392 Decimal expansion of the number x satisfying 3x=exp(-x).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Dec 18 2011

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

			x=0.257627653049736704282916201626097790909692...

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

Cf. A202322.

u = 3; 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, .25, .26}, WorkingPrecision -> 110]
RealDigits[r]  (* A202392 *)
(* other program *)
RealDigits[ ProductLog[1/3], 10, 99] // First (* Jean-François Alcover, Feb 14 2013 *)

lambertw(1/3) \\ G. C. Greubel, Jun 10 2017

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A202320 Decimal expansion of x < 0 satisfying x + 2 = e^x, negated.

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

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A202356 Decimal expansion of the number x satisfying 2x=exp(-x).

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A202355 Decimal expansion of the number x satisfying x-1=exp(-x).

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A202323 Decimal expansion of the number x satisfying x+3=exp(-x).

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A202353 Decimal expansion of the number x satisfying 2*x + 2 = exp(-x), negated.

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Mathematica

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A202354 Decimal expansion of the number x satisfying x+e=exp(-x).

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