

A073243


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


9



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
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OFFSET

0,1


COMMENTS

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
Original definition: Limit of (1/Pi)^...^(1/Pi), n times, as n approaches infinity. Equals exp(LambertW(log(Pi))).
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. See the Mathematica program.  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


LINKS

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) 151164; see the Appendix.
E. Weisstein, Power Tower


FORMULA

x = LambertW(log(Pi))/log(Pi), solution to Pi^x=1/x.  M. F. Hasler, Nov 01 2011


EXAMPLE

0.53934349886230120806079568445...


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 *)


PROG

(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 1E200 specified here) */ n=0; s=1/Pi; t=1; while(abs(ts)>1E200, t=s; s=(1/Pi)^s; n++); print(n, ", ", s)
(PARI) solve(x=0, 1, x1/Pi^x) \\  M. F. Hasler, Nov 01 2011


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).
Sequence in context: A159275 A059031 A245516 * A134943 A105372 A107449
Adjacent sequences: A073240 A073241 A073242 * A073244 A073245 A073246


KEYWORD

cons,nonn


AUTHOR

Rick L. Shepherd, Jul 28 2002


STATUS

approved



