A030797 -id:A030797 - cp's OEIS frontend

A133930 Decimal expansion of the solution to x^x = Phi (A001622, the Golden Ratio).

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

Offset: 1

Rick L. Shepherd, Sep 29 2007

			x = 1.40757981445881000487...

Cf. A001622, A030797.

x=GoldenRatio; RealDigits[Log[x]/ProductLog[Log[x]],10,6! ][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 11 2010 *)

PARI
```
solve(x=1,2,x^x-(1+sqrt(5))/2)
```

A387131 Decimal expansion of the real part of the complex solutions to log(z) = -1/z on the principal branch of log(z).

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Aug 17 2025

			0.16837637908722291055702904019641979887737474829...

Index entries for transcendental numbers.

Cf. A030797, A059526, A059527.

Cf. A387132 (absolute value of the imaginary part).

RealDigits[Re[-1/ProductLog[-1]],10,100][[1]]

Equals Re(-1/LambertW(-1)).

A387132 Decimal expansion of the absolute value of the imaginary part of the complex solutions to log(z) = -1/z on the principal branch of log(z).

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Aug 17 2025

			0.7077541887847276164728458930765162394938408142...

Index entries for transcendental numbers.

Cf. A030797, A059526, A059527.

Cf. A387131 (real part).

RealDigits[Im[-1/ProductLog[-1]],10,100][[1]]

Equals Im(-1/LambertW(-1)).

Equals -Im(-1/LambertW(-1, -1)).

A173169 Decimal expansion of the solution x to x^x = A, the Glaisher-Kinkelin constant (A074962).

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 11 2010

			1.22512630432118191..^1.22512630432118191.. = 1.28242712910062263687534256886979..

Cf. A030437, A030797, A030798, A090625, A133930, A173158-A173166.

x=Glaisher;RealDigits[Log[x]/ProductLog[Log[x]],10,4*5! ][[1]]

(x->x/lambertw(x))(1/12-zeta'(-1)) \\ Charles R Greathouse IV, Dec 12 2013

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

A181171 Decimal expansion of the base x for which the double logarithm of 2 equals the natural log of 2, that is, log_x log_x 2 = log 2.

Original entry on oeis.org

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

Offset: 1

Geoffrey Caveney, Oct 08 2010

			From _R. J. Mathar_, Oct 09 2010: (Start)
1.63662620778092377066392348972183502182...
log_(1.63662..)(2) = 1.4070142427036...
log_(1.63662..)(1.407014..) = A002162. (End)

Cf. A030797, which is the decimal expansion of the base n for which the double logarithm of e (log_n log_n e) = log e = 1, and which is the inverse of LambertW(1).

f := log(log(2))/log(x)-log(log(x))/log(x)-log(2) ; fz := x-f/diff(f,x) ; z := 1.6 ; Digits := 120 ; for i from 1 to 10 do z := evalf(subs(x=z,fz)) ; print(%) ; end do: # R. J. Mathar, Oct 09 2010

RealDigits[ Exp[ ProductLog[Log[2]^2] / Log[2]], 10, 105][[1]] (* Jean-François Alcover, Jan 28 2014 *)

More digits from R. J. Mathar, Oct 09 2010

A332915 Decimal expansion of the constant W(1) + 1/W(1), where W is Lambert's function.

Original entry on oeis.org

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

Offset: 1

Martin Renner, Mar 02 2020

The graph of the exponential function exp(x) moved to the right by W(1) + 1/W(1) touches the graph of the natural logarithm log(x) at point (x,y) = (1/W(1), W(1)) = (A030797, A030178).

			2.33036612476168058322517043916206263018983377385398...

Cf. A030178, A030797, A052888.

evalf[200](LambertW(1) + 1/LambertW(1));

RealDigits[N[LambertW[1] + 1/LambertW[1], 120]][[1]] (* Vaclav Kotesovec, Mar 02 2020 *)

my(x=lambertw(1)); x+1/x \\ Michel Marcus, Mar 02 2020

Equals A030178 + A030797 = A030178 + 1/A030178 = 1/A030797 + A030797.

Equals 2 + Integral_{x=0..1} W(x) dx. - Amiram Eldar, Jul 18 2021

cp's OEIS Frontend

A133930 Decimal expansion of the solution to x^x = Phi (A001622, the Golden Ratio).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

A387131 Decimal expansion of the real part of the complex solutions to log(z) = -1/z on the principal branch of log(z).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A387132 Decimal expansion of the absolute value of the imaginary part of the complex solutions to log(z) = -1/z on the principal branch of log(z).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A173169 Decimal expansion of the solution x to x^x = A, the Glaisher-Kinkelin constant (A074962).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Extensions

A181171 Decimal expansion of the base x for which the double logarithm of 2 equals the natural log of 2, that is, log_x log_x 2 = log 2.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

Extensions

A332915 Decimal expansion of the constant W(1) + 1/W(1), where W is Lambert's function.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

PARI

Formula