A202351 -id:A202351 - 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

A202352 Decimal expansion of greatest x satisfying 3*x = exp(x).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Dec 17 2011

See A202320 for a guide to related sequences. The Mathematica program includes a graph.

			least:  0.61906128673594511215232699402092223330147...
greatest:  1.51213455165784247389673967807203870460...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for transcendental numbers

Cf. A202320.

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 *)
RealDigits[ -ProductLog[-1, -1/3], 10, 99] // First (* Jean-François Alcover, Feb 27 2013 *)

solve(x=1, 2, 3*x-exp(x)) \\ Michel Marcus, Nov 09 2017

Equals -LambertW(-1,-1/3). - Gleb Koloskov, Jun 12 2021

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

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

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A202352 Decimal expansion of greatest x satisfying 3*x = exp(x).

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

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