A030797 -id:A030797 - cp's OEIS frontend

A030178 Decimal expansion of LambertW(1): the solution to x*exp(x) = 1.

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

Offset: 0

N. J. A. Sloane

Sometimes called the Omega constant.
Infinite power tower for c = 1/E, i.e., c^c^c^..., where c = 1/A068985. - Stanislav Sykora, Nov 03 2013
Notice the narrow interval exp(-gamma) < w(1) < gamma, with gamma = A001620. - Jean-François Alcover, Dec 18 2013
Also the solution to x = -log(x). - Robert G. Wilson v, Feb 22 2014

			0.5671432904097838729999686622103555497538157871865125081351310792230457930866...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.11, p. 449.

G. C. Greubel and Stanislav Sykora, Table of n, a(n) for n = 0..10000 (terms 0..1999 from Stanislav Sykora)
Mateus Araújo, Andrew J. P. Garner, and Miguel Navascues, Non-commutative optimization problems with differential constraints, arXiv:2408.02572 [quant-ph], 2024. See p. 13.
blackpenredpen, Finding Omega, featuring Newton's method, video (2018).
Daniel Cummerow, Sound of Mathematics, Constants.
István Mező, An integral representation for the Lambert W function, arXiv:2012.02480 [math.CA], 2020.
Simon Plouffe, Lambert W(1, 0).
Simon Plouffe, The omega constant or W(1).
Stanislav Sykora, Fixed points of the mappings exp(z) and -exp(z) in C, Stan's Library, Vol. VI, 2016.
Eric Weisstein's World of Mathematics, Omega Constant.
Eric Weisstein's World of Mathematics, Lambert W-Function.
Wikipedia, Omega constant.
Wadim Zudilin, Diophantine problems related to the Omega constant, arXiv:2004.11029 [math.NT], 2020.
Index entries for transcendental numbers.

Cf. A019474, A059526, A059527, A238274.

Cf. A276759 (another fixed point of -exp(z)).

Maple
```
evalf(LambertW(1));
```

RealDigits[ ProductLog[1], 10, 111][[1]] (* Robert G. Wilson v, May 19 2004 *)

solve(x=0,1,x*exp(x)-1) \\ Charles R Greathouse IV, Mar 20 2012

lambertw(1) \\ Stanislav Sykora, Nov 03 2013

Equals 1/A030797.

Equals (1/Pi) * Integral_{x=0..Pi} log(1 + sin(x)*exp(x*cot(x))/x) dx (Mező, 2020). - Amiram Eldar, Jul 04 2021

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

Original entry on oeis.org

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

A173159 Decimal expansion of the constant x which satisfies x^x = 5.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 11 2010

			2.12937248...^2.12937248... = 5.
2.12937248..*log(2.12937248..) = 1.609437... = A016628.

Index entries for transcendental numbers

Cf. A030437, A030797, A030798, A173158.

Digits := 20 ; fsolve(x^x=5) ; # R. J. Mathar, Mar 11 2010

x=5;RealDigits[Log[x]/ProductLog[Log[x]],10,6! ][[1]]

log(5)/lambertw(log(5)) \\ Charles R Greathouse IV, Jul 14 2020

Log(5)/W(log(5)).

Keyword:cons added by R. J. Mathar, Mar 11 2010

A173160 Decimal expansion of the constant x satisfying x^x = 6.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 11 2010

			2.2318286..^2.2318286..=6. 2.2318286..*log(2.2318286..) = A016629.

Cf. A030437, A030797, A030798, A173158, A173159.

x=6;RealDigits[Log[x]/ProductLog[Log[x]],10,6! ][[1]]

Digits of log(6)/W(log(6)).

Keyword:cons added by R. J. Mathar, Mar 14 2010

A173161 Decimal expansion of the solution x to x^x=7.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 11 2010

			2.3164549..^2.3164549..=7

Cf. A030437, A030797, A030798, A173158, A173159, A173160.

x=7;RealDigits[Log[x]/ProductLog[Log[x]],10,6! ][[1]]

Digits of log(7)/W(log(7)).

Keyword:cons added by R. J. Mathar, Feb 13 2010

A173162 Decimal expansion of the solution x to x^x=8.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 11 2010

			2.3884234..^2.3884234..=8

Cf. A030437, A030797, A030798, A173158-A173161.

x=8;RealDigits[Log[x]/ProductLog[Log[x]],10,6! ][[1]]

Digits of log(8)/W(log(8)).

Keyword:cons added by R. J. Mathar, Feb 13 2010

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.

A173163 Decimal expansion of the solution x to x^x=9.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 11 2010

			2.450953928..^2.450953928..=9

Index entries for transcendental numbers.

Cf. A030437, A030797, A030798, A173158-A173162.

x=9;RealDigits[Log[x]/ProductLog[Log[x]],10,6! ][[1]]

Digits of log(9)/W(log(9)).

Keyword:cons added by R. J. Mathar, Feb 13 2010

A342360 Decimal expansion of 1/(Omega+1)^2, where Omega=LambertW(1) is the Omega constant.

Original entry on oeis.org

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

Offset: 0

Gleb Koloskov, Mar 09 2021

			0.40717638729656715790289020473539767731...

Cf. A342359, A342361, A030178, A030797.

Omega=LambertW[1]; xi=ArcTan[Sqrt[Omega]]; N[Cos[xi]^4,120]
Omega=LambertW[1]; N[1/(Omega+1)^2,120]
Omega=LambertW[1]; omega=1/Omega; NIntegrate[(-t/LambertW[-1,-t*Omega^omega])^Omega,{t,0,1}, WorkingPrecision->120]

PARI
```
cos(atan(sqrt(lambertw(1))))^4
```
PARI
```
my(Omega=lambertw(1)); 1/(Omega+1)^2
```

Equals cos(A342359)^4 = 1/(A030178+1)^2 = (1-sqrt(A342361))^2.
Equals Integral_{t=0..1} (-t/LambertW(-1,-t*Omega^omega))^Omega, where omega=1/Omega=1/LambertW(1).
Equals A115287^2. - Vaclav Kotesovec, Mar 12 2021

A342361 Decimal expansion of 1/(omega+1)^2, where omega=1/LambertW(1).

Original entry on oeis.org

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

Offset: 0

Gleb Koloskov, Mar 09 2021

			0.1309689005663456008580754336956370484226429615564731843

Cf. A342359, A342360, A030178, A030797.

Omega=LambertW[1]; xi=ArcTan[Sqrt[Omega]]; N[Sin[xi]^4,120]
omega=1/LambertW[1]; N[1/(omega+1)^2,120]
Omega=LambertW[1]; omega=1/Omega; NIntegrate[(-t/LambertW[-1,-t*Omega^omega])^omega,{t,0,1}, WorkingPrecision->120]
RealDigits[1/(1/LambertW[1]+1)^2,10,120][[1]] (* Harvey P. Dale, Mar 26 2025 *)

my(Omega=lambertw(1), xi=atan(sqrt(Omega))); sin(xi)^4

PARI
```
1/(1/lambertw(1)+1)^2
```

Equals Integral_{t=0..1} (-t/W(-1,-t*Omega^omega))^omega, where omega = 1/Omega = 1/LambertW(1).

Equals sin(A342359)^4 = 1/(A030797+1)^2 = (1-sqrt(A342360))^2.

cp's OEIS Frontend

A030178 Decimal expansion of LambertW(1): the solution to x*exp(x) = 1.

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

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A173159 Decimal expansion of the constant x which satisfies x^x = 5.

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A173160 Decimal expansion of the constant x satisfying x^x = 6.

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A173161 Decimal expansion of the solution x to x^x=7.

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A173162 Decimal expansion of the solution x to x^x=8.

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

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