A030178 -id:A030178 - cp's OEIS frontend

A299614 Decimal expansion of e^(2A030178) = e^(2W(1)) = (1/W(1))^2, where W is the Lambert W function (or PowerLog); see Comments.

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

Offset: 1

Clark Kimberling, Mar 01 2018

The Lambert W function satisfies the functional equation e^(W(x) + W(y)) = x*y/(W(x)*W(y)) for x and y greater than -1/e, so that e^(2*W(1)) = (W(1))^(-2). See A299613 for a guide to related constants.

			e^(2*W(1)) = 3.1089547635799361854809454054...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Lambert W-Function

Cf. A299613.

w[x_] := ProductLog[x]; x = 1; y = 1;
N[E^(w[x] + w[y]), 130]   (* A299614 *)
RealDigits[(1/LambertW[1])^2, 10, 100][[1]] (* G. C. Greubel, Mar 03 2018 *)

(1/lambertw(1))^2 \\ G. C. Greubel, Mar 03 2018

A370490 The denominators of a series that converges to the Omega constant (A030178) obtained using Whittaker's root series formula.

Original entry on oeis.org

2, 14, 259, 9657, 200187, 18671081, 7313976065, 1273374259615, 285038137030769, 79755360301275363, 9091712937155442435, 149243024021521700285, 1085736156475373087072485, 3071709182054627484879798019, 2005459027715242401528647218817, 1496371535371115486607560677791759

Offset: 1

Raul Prisacariu, Feb 19 2024

nonn

Whittaker's root series formula is applied to 1 - 2x + x^2/2! - x^3/3! + x^4/4! - x^5/5! + x^6/6! - ..., which is the Taylor expansion of -x + e^(-x). We obtain the following infinite series that converges to the Omega constant (LambertW(1)): LambertW(1) = 1/2 + 1/14 - 1/259 - 5/9657 + 19/200187 - 3/18671081 ... . The sequence is formed by the denominators of the infinite series.

			a(1) is the denominator of -1/-2 = 1/2.
a(2) is the denominator of -(1/2)/((-2)*det ToeplitzMatrix((-2,1),(-2,1/2!))) = -(1/2)/((-2)*(7/2)) = 1/14.
a(3) is the denominator of -det ToeplitzMatrix((1/2!,-2),(1/2!,-1/3!))/(det ToeplitzMatrix((-2,1),(-2,1/2!))*det ToeplitzMatrix((-2,1,0),(-2,1/2!,-1/3!))) = -(-1/12)/((7/2)*(-37/6)) = -1/259.

E. T. Whittaker and G. Robinson, The Calculus of Observations, London: Blackie & Son, Ltd. 1924, pp. 120-123.

Cf. A030178, A370491 (numerator).

for n>1, a(n) is the denominator of the simplified fraction -det ToeplitzMatrix((c(2),c(1),c(0),0,0,...,0),(c(2),c(3),c(4),...,c(n)))/(det ToeplitzMatrix((c(1),c(0),0,...,0),(c(1),c(2),c(3),...,c(n-1)))*det ToeplitzMatrix((c(1),c(0),0,...,0),(c(1),c(2),c(3),...,c(n)))), where c(0)=1, c(1)=-2, c(n) = (-1)^n/n!.

a(9)-a(16) from Chai Wah Wu, Mar 23 2024

A370491 The numerators of a series that converges to the Omega constant (A030178) obtained using Whittaker's root series formula.

Original entry on oeis.org

1, 1, -1, -5, 19, -3, -10187, 146847, 3268961, -211632497, 393324007, 5402916117, -3884618921299, -774402304798329, 148294948981707557, -3311395903665985169, -43463254022673425965, 14469962812566878696039, 6554498075974546253080309, -3074689522272735111427973673

Offset: 1

Raul Prisacariu, Feb 19 2024

sign

Whittaker's root series formula is applied to 1 - 2x + x^2/2! - x^3/3! + x^4/4! - x^5/5! + x^6/6! - ..., which is the Taylor expansion of -x + e^(-x). We obtain the following infinite series that converges to the Omega constant (LambertW(1)): LambertW(1) = 1/2 + 1/14 - 1/259 - 5/9657 + 19/200187 - 3/18671081 ... . The sequence is formed by the numerators of the infinite series.

			a(1) is the numerator of -1/-2 = 1/2.
a(2) is the numerator of -(1/2)/((-2)*det ToeplitzMatrix((-2,1),(-2,1/2!))) = -(1/2)/((-2)*(7/2)) = 1/14.
a(3) is the numerator of -det ToeplitzMatrix((1/2!,-2),(1/2!,-1/3!))/(det ToeplitzMatrix((-2,1),(-2,1/2!))*det ToeplitzMatrix((-2,1,0),(-2,1/2!,-1/3!))) = -(-1/12)/((7/2)*(-37/6)) = -1/259.

E. T. Whittaker and G. Robinson, The Calculus of Observations, London: Blackie & Son, Ltd. 1924, pp. 120-123.

Cf. A030178, A370490 (denominator).

For n > 1, a(n) is the numerator of the simplified fraction -det ToeplitzMatrix((c(2),c(1),c(0),0,0,...,0),(c(2),c(3),c(4),...,c(n)))/(det ToeplitzMatrix((c(1),c(0),0,...,0),(c(1),c(2),c(3),...,c(n-1)))*det ToeplitzMatrix((c(1),c(0),0,...,0),(c(1),c(2),c(3),...,c(n)))), where c(0)=1, c(1)=-2, c(n) = (-1)^n/n!.

a(9)-a(20) from Chai Wah Wu, Mar 23 2024

A073226 Decimal expansion of e^e.

Original entry on oeis.org

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

Offset: 2

Rick L. Shepherd, Jul 21 2002

Given z > 0, there exist positive real numbers x < y, with x^y = y^x = z, if and only if z > e^e. In that case, 1 < x < e < y and (x, y) = ((1 + 1/t)^t, (1 + 1/t)^(t+1)) for some t > 0. (For example, t = 1 gives 2^4 = 4^2 = 16 > e^e.) Proofs of these classical results and applications of them are in Marques and Sondow (2010).
e^e = lim_{n->infinity} ((n+1)/n)^((n+1)^(n+1)/n^n), n > 0 an integer; cf. [Vernescu] wherein it is also stated that the assertions of the previous comment above were proved by Alexandru Lupas in 2006. - L. Edson Jeffery, Sep 18 2012
A weak form of Schanuel's Conjecture implies that e^e is transcendental--see Marques and Sondow (2012).

			15.15426224147926418976043027262991190552854853685613976914...

Harry J. Smith, Table of n, a(n) for n = 2..20000
D. Marques and J. Sondow, The Schanuel Subset Conjecture implies Gelfond's Power Tower Conjecture, arXiv:1212.6931 [math.NT], 2012-2013.
Simon Plouffe, exp(E) to 2000 places
J. Sondow and D. Marques, Algebraic and transcendental solutions of some exponential equations, Annales Mathematicae et Informaticae 37 (2010) 151-164.
A. Vernescu, About the use of a result of Professor Alexandru Lupas to obtain some properties in the theory of the number e, Gen. Math., Vol. 15, No. 1 (2007), 75-80.

Cf. A073233 (Pi^Pi), A049006 (i^i), A001113 (e), A073227 (e^e^e), A004002 (Benford numbers), A056072 (floor(e^e^...^e), n e's), A072364 ((1/e)^(1/e)), A030178 (limit of (1/e)^(1/e)^...^(1/e)), A073229 (e^(1/e)), A073230 ((1/e)^e).

Exp(Exp(1)); // G. C. Greubel, May 29 2018

Mathematica
```
RealDigits[ E^E, 10, 110] [[1]]
```
PARI
```
exp(exp(1))
```

{ default(realprecision, 20080); x=exp(1)^exp(1)/10; for (n=2, 20000, d=floor(x); x=(x-d)*10; write("b073226.txt", n, " ", d)); } \\ Harry J. Smith, Apr 30 2009

Equals Sum_{n>=0} e^n/n!. - Richard R. Forberg, Dec 29 2013

Equals Product_{n>=0} e^(1/n!). - Amiram Eldar, Jun 29 2020

A072597 Expansion of 1/(exp(-x) - x) as exponential generating function.

Original entry on oeis.org

1, 2, 7, 37, 261, 2301, 24343, 300455, 4238153, 67255273, 1185860331, 23000296155, 486655768525, 11155073325917, 275364320099807, 7282929854486431, 205462851526617489, 6158705454187353297, 195465061563672788947, 6548320737474275229347, 230922973019493881984021

Offset: 0

Michael Somos, Jun 23 2002

Polynomials from A140749/A141412 are linked to Stirling1 (see A048594, A129841, A140749). See also P. Flajolet, X. Gourdon, B. Salvy in, available on Internet, RR-1857.pdf (preprint of unavailable Gazette des Mathematiciens 55, 1993, pp. 67-78; for graph 2 see also X. Gourdon RR-1852.pdf, pp. 64-65). What is the corresponding graph for A152650/A152656 = simplified A009998/A119502 linked, via A152818, to a(n), then Stirling2? - Paul Curtz, Dec 16 2008
Denominators in rational approximations of Lambert W(1). See Ramanujan, Notebooks, volume 2, page 22: "2. If e^{-x} = x, shew that the convergents to x are 1/2, 4/7, 21/37, 148/261, &c." Numerators in A006153. - Michael Somos, Jan 21 2019
Call an element g in a semigroup a group element if g^j = g for some j > 1. Then a(n) is the number of group elements in the semigroup of partial transformations of an n-set. Hence a(n) = Sum_{k=0..n} A154372(n,k)*k!. - Geoffrey Critzer, Nov 27 2021

			G.f. = 1 + 2*x + 7*x^2 + 37*x^3 + 261*x^4 + 2301*x^5 + 24343*x^6 + ...

O. Ganyushkin and V Mazorchuk, Classical Finite Transformation Semigroups, Springer, 2009, page 70.
S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 2, see page 22.

Seiichi Manyama, Table of n, a(n) for n = 0..411
W. S. Gray and M. Thitsa, System Interconnections and Combinatorial Integer Sequences, in: System Theory (SSST), 2013 45th Southeastern Symposium on, Date of Conference: 11-11 March 2013.
G. Jiraskova and J. Shallit, The state complexity of star-complement-star, arXiv preprint arXiv:1203.5353 [cs.FL], 2012. - From _N. J. A. Sloane_, Sep 21 2012

Cf. A000110, A006153, A030178, A089148.
Cf. A048993, A052820.
Cf. A006153, A154372.

CoefficientList[Series[1/(Exp[-x]-x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 26 2013 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ 1 / (Exp[-x] - x), {x, 0, n}]]; (* Michael Somos, Jan 21 2019 *)
a[ n_] := If[ n < 0, 0, n! Sum[ (n - k + 1)^k / k!, {k, 0, n}]]; (* Michael Somos, Jan 21 2019 *)

{a(n) = if( n<0, 0, n! * polcoeff( 1 / (exp(-x + x * O(x^n)) - x), n))};

{a(n) = if( n<0, 0, n! * sum(k=0, n, (n-k+1)^k / k!))}; /* Michael Somos, Jan 21 2019 */

E.g.f.: 1 / (exp(-x) - x).
a(n) = n!*Sum_{k=0..n} (n-k+1)^k/k!. - Vladeta Jovovic, Aug 31 2003
a(n) = Sum_{k=0..n} (-1)^(n-k)*Stirling2(n, k)*A052820(k). - Vladeta Jovovic, Apr 12 2004
Recurrence: a(n+1) = 1 + Sum_{j=1..n} binomial(n, j)*a(j)*j. - Jon Perry, Apr 25 2005
E.g.f.: 1/(Q(0) - x) where Q(k) = 1 - x/(2*k+1 - x*(2*k+1)/(x - (2*k+2)/Q(k+1) )); (continued fraction ). - Sergei N. Gladkovskii, Apr 04 2013
a(n) ~ n!/((1+c)*c^(n+1)), where c = A030178 = LambertW(1) = 0.5671432904... - Vaclav Kotesovec, Jun 26 2013
O.g.f.: Sum_{k>=0} k!*x^k/(1 - (k + 1)*x)^(k+1). - Ilya Gutkovskiy, Oct 09 2018
a(n) = A006153(n+1)/(n+1). - Seiichi Manyama, Nov 05 2024

A299613 Decimal expansion of 2*W(1), where W is the Lambert W function (or PowerLog); see Comments.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Mar 01 2018

The Lambert W function satisfies the functional equations
W(x) + W(y) = W(x*y*(1/W(x) + 1/W(y))) = log(x*y)/(W(x)*W(y)) for x and y greater than -1/e, so that 2*W(1) = W(2/W(1)) = -2*log(W(1)).
Guide to related constants:
--------------------------------------------
x        y      W(x) + W(y)  e^(W(x) + W(y))
--------------------------------------------
1        1        A299613        A299614
1        2        A299615        A299616
1        e        A030178        A299617
e        e        2 exactly      e^2 exactly
1        1/e      A299618        A299619
1        3        A299620        A299621
1        1/2      A299622        A299623
1/2      1/2      A126583        A099954
2        2        A299624        A299625
3        3        A299626        A299627
1/3      1/3      A299628        A299629
3/2      3/2      A299630        A299631
e/2      e/2      A299632        A299633

			2*W(1) = 1.13428658081956774599993...

Clark Kimberling, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Lambert W-Function.
Index entries for sequences related to LambertW function.
Index entries for transcendental numbers.

Cf. A299614-A299633.

w[x_] := ProductLog[x]; x = 1; y = 1; u = N[w[x] + w[y], 100]
RealDigits[u, 10][[1]]  (* A299613 *)
RealDigits[2 ProductLog[1], 10, 111][[1]] (* Robert G. Wilson v, Mar 02 2018 *)

2*lambertw(1) \\ G. C. Greubel, Mar 07 2018

Equals 2*A030178.

A030797 Decimal expansion of the constant x such that x^x = e. Inverse of W(1), where W is Lambert's function.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Simon Plouffe

Decimal expansion of the solution to y*log(y) = 1. - Benoit Cloitre, Mar 30 2002
Let u(n+1) = exp(1/u(n)) then for any u(1) which is nonzero and real (positive or negative), lim n -> infinity u(n) = 1.763222834.... - Benoit Cloitre, Aug 06 2002
Conjecture: Another series can be defined as follows. Let z = a + b*i <> 0 be complex, and let z = v^v. Then log(z) + v = v*(1 + log(v)), so f(z, v) = (log(z) + v)/(1 + log(v)) = v. Suppose lim_{n -> infinity} (log(z) + v(n))/(1 + Log(v(n))) = v, for some sequence {v(n)}. Then, since v(n) -> v(n+1), similarly f_(n+1)(z, v) = v(n+1) = (log(z) + v(n))/(1 + log(v(n))). If Im(z) <> 0, recall that log(z) is multi-valued, so one might take both log(z) and log(v(n)) modulo 2*Pi*i. If Im(z) = 0 (i.e., if z is real), then one should use the recurrence f_(n+1)(z, v) = v(n+1) = (log(z) + v(n))/(1 + abs(log(v(n)))). For example, when z = e, we have lim_{n -> infinity} (1 + v(n))/(1 + abs(log(v(n)))) = 1.763222..., for v(0) <> 1/e, with apparent quadratic convergence, and most rapidly when v(0) = 1. Pathologies occur when v(0) is in the vicinity of a fixed point of f(z, v); e.g., if z = 2^(1/4), then such a fixed point is c = 0.806693797003867301..., so f_(n)(z, v) -> c, for all n, with a(0) near c. The constant c was calculated to 250 digits by Joerg Arndt. - L. Edson Jeffery, Apr 12 2011

			1.763222834351896710225201776951707080436017986667473634570456905547275847...

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 448-452.

G. C. Greubel, Table of n, a(n) for n = 1..10000
Simon Plouffe, 1/W(1), the inverse of the omega number:W(1)
Simon Plouffe, Plouffe's Inverter, 1/W(1), the inverse of the omega number:W(1)
Simon Plouffe, Project Gutenberg Etext of Miscellaneous Mathematical Constants #13 in our math series

Cf. A001113, A030178, A299614, A299617.

RealDigits[1/ProductLog[1], 10, 111][[1]] (* Robert G. Wilson v *)
RealDigits[x/.FindRoot[x^x==E,{x,1},WorkingPrecision->120]][[1]] (* Harvey P. Dale, Jun 19 2024 *)

solve(x=1,2,x^x-exp(1)) \\ Charles R Greathouse IV, Apr 01 2012

solve(x=1,2,log(x)*x - 1) \\ John W. Nicholson, Apr 10 2015

1/lambertw(1) \\ G. C. Greubel, Mar 02 2018

Equals 1/A030178.
Equals e^A030178. - Colin Linzer, Nov 20 2024
Equals sqrt(A299614) = A299617/e. - Hugo Pfoertner, Nov 20 2024

Broken URL to Project Gutenberg replaced by Georg Fischer, Jan 03 2009

A052820 Expansion of e.g.f. 1/(1 - x + log(1 - x)).

Original entry on oeis.org

1, 2, 9, 62, 572, 6604, 91526, 1480044, 27353448, 568731648, 13138994112, 333895239072, 9256507508112, 278000959058016, 8991458660924112, 311585506208924064, 11517363473843526912, 452332548042633835776

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Previous name was: A simple grammar.

a(n) is the number of ways to seat n people at circular tables, then linearly order the tables, then designate some (possibly all or none) of the tables at which only one person is seated. a(2) = 9 because we have: (1)(2), (1')(2), (1)(2'), (1')(2'), (2)(1), (2')(1), (2)(1'), (2')(1'), (1,2). Cf. A007840. - Geoffrey Critzer, Nov 05 2013

Seiichi Manyama, Table of n, a(n) for n = 0..395
W. S. Gray and M. Thitsa, System Interconnections and Combinatorial Integer Sequences, in: System Theory (SSST), 2013 45th Southeastern Symposium on, Date of Conference: 11-11 Mar 2013.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 785
Makhin Thitsa and W. Steven Gray, On the Radius of Convergence of Cascaded Analytic Nonlinear Systems, 2011 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC), Orlando, FL, USA, December 12-15, 2011, pp. 3830-3835.
M. Thitsa and W. S. Gray, On the radius of convergence of cascaded analytic nonlinear systems: The SISO case, System Theory (SSST), 2011 IEEE 43rd Southeastern Symposium on, 14-16 March 2011, pp. 30-36.
Makhin Thitsa and W. Steven Gray, On the Radius of Convergence of Interconnected Analytic Nonlinear Input-Output Systems, SIAM Journal on Control and Optimization, Vol. 50, No. 5, 2012, pp. 2786-2813. - From _N. J. A. Sloane_, Dec 26 2012

Cf. A030178.
Cf. A367851, A367852, A367853.
Cf. A367845, A367846, A367847.

spec := [S,{C=Cycle(Z),B=Union(C,Z),S=Sequence(B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

CoefficientList[Series[1/(1-x+Log[1-x]), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 01 2013 *)

E.g.f.: -1/(-1+x+log(-1/(-1+x))).
a(n) ~ n! * (1/(1-LambertW(1)))^n/(1/LambertW(1)-LambertW(1)). - Vaclav Kotesovec, Oct 01 2013
a(0) = 1; a(n) = n * a(n-1) + Sum_{k=0..n-1} binomial(n,k) * (n-k-1)! * a(k). - Ilya Gutkovskiy, Apr 26 2021

New name using e.g.f., Vaclav Kotesovec, Oct 01 2013

A059526 Decimal expansion of real part of solution to z = log z.

Original entry on oeis.org

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

Offset: 0

Fabian Rothelius, Jan 21 2001

Repeatedly take logs, starting from any number not equal to 0, 1, e, e^e, e^(e^e), etc. and you will converge to 0.31813150... + 1.33723570...*I.
A complex number w with a negative imaginary part will converge to the conjugate of z since log(conjugate(w)) = conjugate(log(w)). - Gerald McGarvey, Mar 02 2009
This z and its conjugate are the only two complex solutions of z=log(z) on the principal branch of log(z), and of exp(z)=z for |arg(z)| <= Pi. They are also the only nontrivial (z!=0) principal branch solutions of z=W(z^2), W being the Lambert W-function. Though the two values are iterative attractors of the mapping z->log(z), the convergence is rather slow; the precision improves by slightly more than one binary bit every 2.25 iterations (about 7500 iterations are needed to make stable the first 1000 decimal digits). - Stanislav Sykora, Jun 07 2015

			z = 0.31813150520476413531265425158766451720351761387139986692237... + 1.33723570143068940890116214319371061253950213846051241887631... *i

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Stanislav Sykora, Fixed points of the mappings exp(z) and -exp(z) in C, 2016.
Wolfram Research, FixedPoint

Imaginary part is A059527.
Cf. A030178.
Cf: A277681 (another fixed point of exp(z)).

RealDigits[ Re[ N[ FixedPoint[ Log, 1 + I, 910], 105]]] [[1]]
RealDigits[ N[ Re[ ProductLog[-1]], 105]][[1]] (* Jean-François Alcover, Feb 01 2012 *)
RealDigits[Re[x/.FindRoot[x-Log[x]==0,{x,.5,1},WorkingPrecision->200]],10,120][[1]] (* Harvey P. Dale, Aug 07 2022 *)

z=I;for(k=1,16000,z=log(z));real(z) \\ Stanislav Sykora, Jun 07 2015 \\ Using realprecision \p 2010

z=I; for(k=1, 10, z-=(z-log(z))/(1-1/z)); real(z) \\ Jeremy Tan, Sep 23 2017

More terms from Vladeta Jovovic, Feb 26 2001

Edited and extended by Robert G. Wilson v, Aug 22 2002

A059527 Decimal expansion of imaginary part of solution to z = log z.

Original entry on oeis.org

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

Offset: 1

Fabian Rothelius, Jan 21 2001

Repeatedly take logs, starting from any number not equal to 0, 1, e, e^e, e^(e^e), etc. and you will converge to 0.31813150... + 1.33723570...*I.

			z = 0.31813150520476413531265425158766451720351761387139986692237... + 1.33723570143068940890116214319371061253950213846051241887631... *i.

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.
Wolfram Research, FixedPoint

Real part is A059526.

Cf. A030178.

RealDigits[ Im[ N[ FixedPoint[ Log, 1 + I, 910], 105]]] [[1]]
RealDigits[ N[ Im[ ProductLog[-1]], 105]][[1]] (* Jean-François Alcover, Feb 01 2012 *)

z=I;for(k=1,16000,z=log(z));imag(z)  \\ Using realprecision \p 2010. - Stanislav Sykora, Jun 07 2015

z=I; for(k=1, 10, z-=(z-log(z))/(1-1/z)); imag(z) \\ Jeremy Tan, Sep 23 2017

More terms from Vladeta Jovovic, Feb 26 2001

Edited and extended by Robert G. Wilson v, Aug 22 2002

cp's OEIS Frontend

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A299613 Decimal expansion of 2*W(1), where W is the Lambert W function (or PowerLog); see Comments.

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A299614 Decimal expansion of e^(2A030178) = e^(2W(1)) = (1/W(1))^2, where W is the Lambert W function (or PowerLog); see Comments.