A030178 -id:A030178 - cp's OEIS frontend

A073243 Decimal expansion of exp(-LambertW(log(Pi))), solution to x = 1/Pi^x.

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

Offset: 0

Rick L. Shepherd, Jul 28 2002

Original definition: Limit of (1/Pi)^...^(1/Pi), n times, as n approaches infinity. Equals exp(-LambertW(log(Pi))).
The value can be obtained by iterating x -> 1/Pi^x with any real starting value, but convergence is linear and slow: about 5 iterations are needed for each additional decimal digit. - M. F. Hasler, Nov 01 2011
According to the Weisstein link, infinite iterated exponentiation such as used here, which is referred to both as an "infinite power tower" and "h(x)" -- with graph and other notations -- "converges iff e^(-e) <= x <= e^(1/e) as shown by Euler (1783) and Eisenstein (1844)" (citing Le Lionnais and Wells references). e^(-e) = A073230. e^(1/e) = A073229. x of interest here = 1/Pi = A049541. (1/A073243)^(1/A073243) = A030437^A030437 = Pi.
If y = h(x) = x^x^x^... converges, then by substitution y = x^y. So x^x^x^... is a solution y to the equation y^(1/y) = x. - Jonathan Sondow, Aug 27 2011
The expressions involving "..." in the above comment are misleading, since the limit is not obtained by applying additional "^x" to the previous expression, i.e., iterating "t -> t^x", but corresponds to iterations of "t -> x^t". - M. F. Hasler, Nov 01 2011

			0.53934349886230120806079568445...

Stanislav Sykora, Table of n, a(n) for n = 0..1999
J. Sondow and D. Marques, Algebraic and transcendental solutions of some exponential equations, Annales Mathematicae et Informaticae 37 (2010) 151-164; see the Appendix, arXiv:1108.6096.
Eric Weisstein's World of Mathematics, Power Tower

Cf. A000796 (Pi), A049541 (1/Pi), A073240 ((1/Pi)^(1/Pi)), A073241 ((1/Pi)^(1/Pi)^(1/Pi)), A030437 (reciprocal of A073243), A030178 (corresponding limit for 1/e), A030797 (reciprocal of A030178).

y /. FindRoot[y^(1/y) == 1/Pi, {y, 1}, WorkingPrecision -> 100] (* Jonathan Sondow, Aug 27 2011 *)
First[RealDigits[Exp[-ProductLog[Log[Pi]]], 10, 104]] (* Vladimir Reshetnikov, Nov 01 2011 *)

/* The program below was run with precision set to 1000 digits */ /* n is the number of iterated exponentiations performed. */ /* (n turns out to be 954 with 1E-200 specified here) */ n=0; s=1/Pi; t=1; while(abs(t-s)>1E-200, t=s; s=(1/Pi)^s; n++); print(n,",",s)

solve(x=0,1,x-1/Pi^x)  \\ M. F. Hasler, Nov 01 2011

x = LambertW(log(Pi))/log(Pi), solution to Pi^x=1/x. - M. F. Hasler, Nov 01 2011

A238274 Decimal expansion of abs(LambertW(-1)).

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Feb 24 2014

			1.37455701074370748653...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Stanislav Sykora, Fixed points of the mappings exp(z) and -exp(z) in C, 2016.

Cf. A009306, A030178, A177380, A185221.

RealDigits[N[Abs[LambertW[-1]], 105]][[1]]

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

A196515 Decimal expansion of the number x satisfying x*e^x = 2.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Oct 03 2011

			0.852605502013725491346472414695317466898...

G. C. Greubel, Table of n, a(n) for n = 0..5000
Eric Weisstein's World of Mathematics, Lambert W-Function
Index entries for transcendental numbers

Plot[{E^x, 1/x, 2/x, 3/x, 4/x}, {x, 0, 2}]
t = x /. FindRoot[E^x == 1/x, {x, 0.5, 1}, WorkingPrecision -> 100]
RealDigits[t]  (* A030178 *)
t = x /. FindRoot[E^x == 2/x, {x, 0.5, 1}, WorkingPrecision -> 100]
RealDigits[t]  (* A196515 *)
t = x /. FindRoot[E^x == 3/x, {x, 0.5, 2}, WorkingPrecision -> 100]
RealDigits[t]  (* A196516 *)
t = x /. FindRoot[E^x == 4/x, {x, 0.5, 2}, WorkingPrecision -> 100]
RealDigits[t]  (* A196517 *)
t = x /. FindRoot[E^x == 5/x, {x, 0.5, 2}, WorkingPrecision -> 100]
RealDigits[t]  (* A196518 *)
t = x /. FindRoot[E^x == 6/x, {x, 0.5, 2}, WorkingPrecision -> 100]
RealDigits[t]  (* A196519 *)
RealDigits[ ProductLog[2], 10, 100] // First (* Jean-François Alcover, Feb 26 2013 *)
(* A good approximation (the first 30 digits) is given by this power series evaluated at z=2, expanded at log(z): *)
Clear[x, a, nn, b, z]
z = 2;
nn = 100;
a = Series[Exp[-x], {x, N[Log[z], 50], nn}];
b = Normal[InverseSeries[Series[x/a, {x, 0, nn}]]];
x = z;
N[b, 30]
N[LambertW[z], 30] (* Mats Granvik, Nov 29 2013 *)
RealDigits[LambertW[2], 10, 50][[1]] (* G. C. Greubel, Nov 16 2017 *)

lambertw(2) \\ G. C. Greubel, Nov 16 2017

From A.H.M. Smeets, Nov 19 2018: (Start)
Equals LambertW(2).
Consider LambertW(z), where z is a complex number: let x(0) be an arbitrary complex number; x(n+1) = z*exp(-x(n)); if lim_{n -> inf} x(n) exists (which is the case for z = 2), then LambertW(z) = lim_{n -> inf} x(n). The region in the complex plane for which this seems to work is as follows: let z = x+iy, then -1/e < x < e for y = 0 and -c < y < c, c = 1.9612... for x = 0. It is not known if the area is open or closed. (End)

A276759 Decimal expansion of the real part of the fixed point of -exp(z) in C congruent with the branch K=1 of log(z)+2PiK*i.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Nov 12 2016

The negated exponential mapping -exp(z) has in C a denumerable set of fixed points z_k with even k, which are the solutions of exp(z)+z = 0. The solutions with positive and negative indices k form mutually conjugate pairs, such as this z_2 and z_-2. A similar situation arises also for the fixed points of the mapping +exp(z). My link explains why is it convenient to use even indices for the fixed points of -exp(z) and odd ones for those of +exp(z). Setting K = sign(k)*floor(|k|/2), an even-indexed z_k is also a solution of z = log(-z)+2*Pi*K*i. Moreover, an even-indexed z_k equals -W_L(1), where W_L is the L-th branch of the Lambert W function, with L=-floor((k+1)/2). For any nonzero K, the mapping M_K(z) = log(-z)+2*Pi*K*i has the even-indexed z_k as its unique attractor, convergent from any nonzero point in C (the case K=0 is an exception, discussed in my linked document).

The value listed here is the real part of z_2 = a + i*A276760.

			1.533913319793574507919741082072733779785298610650766671733076...

Stanislav Sykora, Table of n, a(n) for n = 1..2000
S. Sykora, Fixed points of the mappings exp(z) and -exp(z) in C, Stan's Library, Vol.VI, Oct 2016.
Eric Weisstein's World of Mathematics, Exponential Function.
Wikipedia, Exponential function.

Fixed points of -exp(z): z_0: A030178 (real-valued), and z_2: A276760 (imaginary part), A276761 (modulus).

Fixed points of +exp(z): z_1: A059526, A059527, A238274, and z_3: A277681, A277682, A277683.

RealDigits[Re[-ProductLog[-1, 1]], 10, 105][[1]] (* Jean-François Alcover, Nov 12 2016 *)

default(realprecision,2050);eps=5.0*10^(default(realprecision))
M(z,K)=log(-z)+2*Pi*K*I; \\ the convergent mapping (any K!=0)
K=1;z=1+I;zlast=z;
while(1,z=M(z,K);if(abs(z-zlast)

Let z_2 = A276759+i*A276760. Then z_2 = -exp(z_2) = log(-z_2)+2*Pi*i = -W_-1(1).

A277681 Decimal expansion of the real part of the fixed point of exp(z) in C congruent with the branch K=1 of log(z)+2PiK*i.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Nov 12 2016

The exponential mapping exp(z) has in C a denumerable set of fixed points z_k with odd k, which are the solutions of exp(z) = z. The solutions with positive and negative indices k form mutually conjugate pairs, such as z_3 and z_-3. A similar situation arises also for the related fixed points of the mapping -exp(z). My link explains why is it convenient to use odd indices for the fixed points of +exp(z) and even indices for those of -exp(z). Setting K = sign(k)*floor(|k|/2), an odd-indexed z_k is also a fixed point of the logarithmic function in its K-th branch, i.e., a solution of z = log(z)+2*Pi*K*i. Moreover, an odd-indexed z_k equals -W_L(-1), where W_L is the L-th branch of the Lambert W function, with L = -floor((k+1)/2). For any K, the mapping M_K(z) = log(z)+2*Pi*K*i has z_k as its unique attractor, convergent from any nonzero point in C (an exception occurs for K=0, for which M_0(z) has two attractors, z_1 and z_-1, as described in my linked document).

The value listed here is the real part of z_3 = a + i*A277682.

			2.062277729598283884978486720008045951283592306704591613100984...

Stanislav Sykora, Table of n, a(n) for n = 1..2000
S. Sykora, Fixed points of the mappings exp(z) and -exp(z) in C, Stan's Library, Vol.VI, Oct 2016.
Eric Weisstein's World of Mathematics, Exponential Function.
Wikipedia, Exponential function.

Fixed points of +exp(z): z_1: A059526, A059527, A238274, and z_3: A277682 (imaginary part), A277683 (modulus).

Fixed points of -exp(z): z_0: A030178, and z_2: A276759, A276760, A276761.

RealDigits[Re[-ProductLog[-2, -1]], 10, 105][[1]] (* Jean-François Alcover, Nov 12 2016 *)

default(realprecision,2050);eps=5.0*10^(default(realprecision))
M(z,K)=log(z)+2*Pi*K*I; \\ the convergent mapping (any K)
K=1;z=1+I;zlast=z;
while(1,z=M(z,K);if(abs(z-zlast)

Let z_3 = A277681+i*A277682. Then z_3 = exp(z_3) = log(z_3)+2*Pi*i = -W_-2(-1).

A038458 Decimal expansion of the solution to 127^x - 113^x = 1. This is the smallest x such that q^x - p^x = 1 for two successive primes p, q.

Original entry on oeis.org

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

Offset: 0

M. I. Petrescu (mipetrescu(AT)yahoo.com)

Generalizes Andrica's conjecture prime(n+1)^(1/2) - prime(n)^(1/2) < 1 to prime(n+1)^c - prime(n)^c < 1 if c is less than this number.
Is this constant rational or irrational? I conjecture it is irrational. - Sukanto Bhattacharya (susant5au(AT)yahoo.com.au), Apr 28 2008
The first five digits are the same as the first five of A030178 = LambertW(1). - John W. Nicholson, Dec 11 2013
Although the description of the sequence defines it as "the smallest x" with a certain property, this is conjectured, not yet proven. Numerical evidence supports the conjecture. - Hal M. Switkay, Jun 02 2021

			0.567148130202017714646846875533482564586790249388638206840285221826806766338276...

Harry J. Smith, Table of n, a(n) for n = 0..20000
Octavian Cira, Smarandache's conjecture on consecutive primes, International J. Math. Combin. 4 (2014), pp. 69-91.
David Lowry-Duda, A short note on gaps between powers of consecutive primes, arXiv:1709.07847 [math.NT], 2017.
M. L. Perez, Five Smarandache conjectures on primes, Arizona State University, Special Collections.
F. Smarandache, Conjectures which generalize Andrica's conjecture, arXiv:0707.2584 [math.GM], 2007; Octogon 7:1 (1999), pp. 173-176.
Eric Weisstein's World of Mathematics, Andrica's Conjecture
Eric Weisstein's World of Mathematics, Smarandache Constants

RealDigits[x/.FindRoot[127^x-113^x==1,{x,0.5},WorkingPrecision->150]][[1]] (* Harvey P. Dale, Oct 24 2017 *)

default(realprecision, 20080); x=solve(x=.5,.6,127^x-113^x-1); d=0; for (n=0, 20000, x=(x-d)*10; d=floor(x); write("b038458.txt", n, " ", d)); \\ Harry J. Smith, Apr 13 2009

Title improved, incorrect formula deleted, and other edits by M. F. Hasler, Jan 02 2015

A265953 E.g.f.: Product_{k>=1} 1/(1 - exp(x)*x^k).

Original entry on oeis.org

1, 1, 6, 39, 328, 3305, 39396, 536053, 8210784, 139670721, 2612934820, 53260680341, 1175587507392, 27929705129521, 710678763809028, 19284199100275845, 555961318128936256, 16972543570002866945, 547046699544108738756, 18566047855851466092949

Offset: 0

Vaclav Kotesovec, Dec 19 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..408

Cf. A030178, A030797, A265952.

nmax=20; CoefficientList[Series[Product[1/(1-E^x*x^k), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!

a(n) ~ c * n! / LambertW(1)^n, where c = 1/(1 + LambertW(1)) * Product_{j>=1} 1/(1 - LambertW(1)^j) = 3.40413121452412914124892504613759007312040569..., LambertW(1) = A030178.

A276760 Decimal expansion of the imaginary part of the fixed point of -exp(z) in C congruent with the branch K=1 of log(z)+2PiK*i.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Nov 12 2016

Imaginary part of the complex constant z_2 whose real part is in A276759 (see the latter entry for more information).

			4.375185153061898385470906564852584291623823114677011864961044...

Stanislav Sykora, Table of n, a(n) for n = 1..2000
Stanislav Sykora, Fixed points of the mappings exp(z) and -exp(z) in C, 2016.

Fixed points of -exp(z): z_0: A030178, and z_2: A276759 (real part), A276761 (modulus).

Fixed points of +exp(z): z_1: A059526, A059527, A238274, and z_3: A277681, A277682, A277683.

RealDigits[Im[-ProductLog[-1, 1]], 10, 105][[1]] (* Jean-François Alcover, Nov 12 2016 *)

default(realprecision,2050);eps=5.0*10^(default(realprecision))
M(z,K)=log(-z)+2*Pi*K*I; \\ the convergent mapping (any K)
K=1;z=1+I;zlast=z;
while(1,z=M(z,K);if(abs(z-zlast)

Let z_2 = A276759+i*A276760. Then z_2 = -exp(z_2) = log(-z_2)+2*Pi*i = -W_-1(1).

A276761 Decimal expansion of the modulus of the fixed point of -exp(z) in C congruent with the branch K=1 of log(z)+2PiK*i.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Nov 12 2016

Modulus of z_2 = A276759+i*A276760. See A276759 for more information.

			4.636284632786625189544952318034205387044699355677575252963935...

Stanislav Sykora, Table of n, a(n) for n = 1..2000
Stanislav Sykora, Fixed points of the mappings exp(z) and -exp(z) in C, 2016.

Fixed points of -exp(z): z_0: A030178, and z_2: A276759 (real part), A276760 (imaginary part).

Fixed points of +exp(z): z_1: A059526, A059527, A238274, and z_3: A277681, A277682, A277683.

RealDigits[Norm[ProductLog[1, 1]], 10, 105][[1]] (* Jean-François Alcover, Nov 12 2016 *)

default(realprecision,2050);eps=5.0*10^(default(realprecision))
M(z,K)=log(-z)+2*Pi*K*I; \\ the convergent mapping (any K)
K=1;z=1+I;zlast=z;
while(1,z=M(z,K);if(abs(z-zlast)

cp's OEIS Frontend

A073243 Decimal expansion of exp(-LambertW(log(Pi))), solution to x = 1/Pi^x.

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A238274 Decimal expansion of abs(LambertW(-1)).

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

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A196515 Decimal expansion of the number x satisfying x*e^x = 2.

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A276759 Decimal expansion of the real part of the fixed point of -exp(z) in C congruent with the branch K=1 of log(z)+2*Pi*K*i.

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A277681 Decimal expansion of the real part of the fixed point of exp(z) in C congruent with the branch K=1 of log(z)+2*Pi*K*i.

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A038458 Decimal expansion of the solution to 127^x - 113^x = 1. This is the smallest x such that q^x - p^x = 1 for two successive primes p, q.

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A276759 Decimal expansion of the real part of the fixed point of -exp(z) in C congruent with the branch K=1 of log(z)+2PiK*i.

A277681 Decimal expansion of the real part of the fixed point of exp(z) in C congruent with the branch K=1 of log(z)+2PiK*i.

A276760 Decimal expansion of the imaginary part of the fixed point of -exp(z) in C congruent with the branch K=1 of log(z)+2PiK*i.

A276761 Decimal expansion of the modulus of the fixed point of -exp(z) in C congruent with the branch K=1 of log(z)+2PiK*i.