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 1-5 of 5 results.

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

A073234 Decimal expansion of Pi^(Pi^Pi).

Original entry on oeis.org

1, 3, 4, 0, 1, 6, 4, 1, 8, 3, 0, 0, 6, 3, 5, 7, 4, 3, 5, 2, 9, 7, 4, 4, 9, 1, 2, 9, 6, 4, 0, 1, 3, 1, 4, 1, 5, 0, 9, 9, 3, 7, 4, 9, 7, 4, 5, 7, 3, 4, 9, 9, 2, 3, 7, 7, 8, 7, 9, 2, 7, 5, 1, 6, 5, 8, 6, 0, 3, 4, 0, 9, 2, 6, 1, 9, 0, 9, 4, 0, 6, 8, 1, 4, 8, 2, 6, 9, 4, 7, 2, 6, 1, 1, 3, 0, 1, 1, 4, 2, 2, 7, 3, 4, 3
Offset: 19

Views

Author

Rick L. Shepherd, Jul 25 2002

Keywords

Examples

			1340164183006357435.29744912964...

References

  • Christopher Creutzig & Walter Oevel, MuPAD Tutorial. Berlin: Springer-Verlag (2004): 339.

Links

Crossrefs

Cf. A000796 (Pi), A073233 (Pi^Pi), A202955 (Pi^Pi^Pi^Pi), A073236 (Pi^Pi^...^Pi, n times, rounded), A073235 ((Pi^Pi)^Pi), A073241 ((1/Pi)^(1/Pi)^(1/Pi)).

Programs

  • Mathematica
    RealDigits[Pi^Pi^Pi, 10, 100][[1]] (* Alonso del Arte, Jul 03 2012 *)
  • MuPAD
    DIGITS := 100:
float(PI^(PI^PI)) // from Creutzig & Oevel (2004)
  • PARI
    Pi^Pi^Pi
  • PARI
    { default(realprecision, 20100); x=Pi^Pi^Pi/10^18; for (n=19, 20000, d=floor(x); x=(x-d)*10; write("b073234.txt", n, " ", d)); } \\ Harry J. Smith, May 01 2009, corrected May 19 2009, had -n

A073240 Decimal expansion of (1/Pi)^(1/Pi).

Original entry on oeis.org

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

Views

Author

Rick L. Shepherd, Jul 27 2002

Keywords

Comments

(1/Pi)^(1/Pi) = Pi^(-1/Pi) = 1/(Pi^(1/Pi)) (reciprocal of A073238).

Examples

			0.69462799224682615312438376141...

Links

Crossrefs

Cf. A000796 (Pi), A049541 (1/Pi), A073241 ((1/Pi)^(1/Pi)^(1/Pi)), A073243 (limit of (1/Pi)^(1/Pi)^...^(1/Pi)), A073238 (Pi^(1/Pi)), A073239 ((1/Pi)^Pi), A073233 (Pi^Pi).

Programs

  • Mathematica
    First[RealDigits[(1/Pi)^(1/Pi),10,100]] (* Paolo Xausa, Nov 07 2023 *)
  • PARI
    (1/Pi)^(1/Pi)

A073235 Decimal expansion of (Pi^Pi)^Pi.

Original entry on oeis.org

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

Views

Author

Rick L. Shepherd, Jul 25 2002

Keywords

Examples

			80662.6659385545967384983601897...

Crossrefs

Cf. A000796 (Pi), A073233 (Pi^Pi), A073234 (Pi^Pi^Pi), A073241 ((1/Pi)^(1/Pi)^(1/Pi)), A073242 (((1/Pi)^(1/Pi))^(1/Pi)).

Programs

  • Mathematica
    RealDigits[(Pi^Pi)^Pi, 10, 105][[1]] (* Alonso del Arte, Jul 03 2012 *)
  • PARI
    (Pi^Pi)^Pi

A073242 Decimal expansion of ((1/Pi)^(1/Pi))^(1/Pi).

Original entry on oeis.org

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

Views

Author

Rick L. Shepherd, Jul 28 2002

Keywords

Examples

			0.89048823592032706709152866456...

Crossrefs

Cf. A000796 (Pi), A049541 (1/Pi), A073240 ((1/Pi)^(1/Pi)), A073241 ((1/Pi)^(1/Pi)^(1/Pi)), A073235 ((Pi^Pi)^Pi), A073234 (Pi^Pi^Pi).

Programs

  • Mathematica
    RealDigits[ N[(1/Pi^(1/Pi))^(1/Pi), 110]][[1]]
  • PARI
    \\ This shorter statement is equivalent to ((1/Pi)^(1/Pi))^(1/Pi): Pi^(-Pi^-2)
Showing 1-5 of 5 results.

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