cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A100947 Continued fraction of the solution of x^x=Pi (A030437).

Original entry on oeis.org

1, 1, 5, 1, 5, 1, 6, 3, 2, 1, 2, 22, 2, 8, 3, 2, 1, 9, 1, 1, 2, 1, 1, 5, 1, 1, 1, 1, 4, 11, 2, 1, 1, 1, 1, 1, 2, 8, 1, 1, 1, 8, 8, 1, 5, 1, 3, 1, 1, 3, 1, 10, 2, 3, 1, 18, 4, 1, 2, 6, 1, 1, 1, 1, 51, 1, 1, 1, 7, 2, 39, 1, 8, 4, 1, 1, 1, 1, 13, 15, 1, 1, 5, 1, 2, 5, 4, 1, 6, 1, 10, 1, 13, 2, 2702, 6, 4, 1, 3
Offset: 0

Views

Author

Mark Hudson (mrmarkhudson(AT)hotmail.com), Nov 23 2004

Keywords

Crossrefs

Cf. A030437.

Programs

  • Mathematica
    ContinuedFraction[x/.FindRoot[x^x==\[Pi],{x,1.8}, WorkingPrecision->100], 100] (* Harvey P. Dale, Sep 07 2011 *)

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

Views

Author

Rick L. Shepherd, Jul 28 2002

Keywords

Comments

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

Examples

			0.53934349886230120806079568445...

Links

Crossrefs

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).

Programs

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

Formula

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

Views

Author

Vladimir Joseph Stephan Orlovsky, Feb 11 2010

Keywords

Examples

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

Links

Crossrefs

Cf. A030437, A030797, A030798, A173158.

Programs

  • Maple
    Digits := 20 ; fsolve(x^x=5) ; # R. J. Mathar, Mar 11 2010
  • Mathematica
    x=5;RealDigits[Log[x]/ProductLog[Log[x]],10,6! ][[1]]
  • PARI
    log(5)/lambertw(log(5)) \\ Charles R Greathouse IV, Jul 14 2020

Formula

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

Extensions

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

Views

Author

Vladimir Joseph Stephan Orlovsky, Feb 11 2010

Keywords

Examples

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

Crossrefs

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

Programs

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

Formula

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

Extensions

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

Views

Author

Vladimir Joseph Stephan Orlovsky, Feb 11 2010

Keywords

Examples

			2.3164549..^2.3164549..=7

Crossrefs

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

Programs

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

Formula

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

Extensions

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

Views

Author

Vladimir Joseph Stephan Orlovsky, Feb 11 2010

Keywords

Examples

			2.3884234..^2.3884234..=8

Crossrefs

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

Programs

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

Formula

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

Extensions

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

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

Views

Author

Vladimir Joseph Stephan Orlovsky, Feb 11 2010

Keywords

Examples

			2.450953928..^2.450953928..=9

Links

Crossrefs

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

Programs

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

Formula

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

Extensions

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

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

Views

Author

Vladimir Joseph Stephan Orlovsky, Feb 11 2010

Keywords

Examples

			1.22512630432118191..^1.22512630432118191.. = 1.28242712910062263687534256886979..

Crossrefs

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

Programs

  • Mathematica
    x=Glaisher;RealDigits[Log[x]/ProductLog[Log[x]],10,4*5! ][[1]]
  • PARI
    (x->x/lambertw(x))(1/12-zeta'(-1)) \\ Charles R Greathouse IV, Dec 12 2013

Extensions

Keyword:cons added by R. J. Mathar, Feb 13 2010
Showing 1-8 of 8 results.

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