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A194346
Decimal expansion of h_o(1/17), where h_o(x) is the odd infinite power tower function.
3
2, 0, 4, 2, 7, 4, 7, 3, 6, 6, 6, 5, 5, 1, 8, 4, 9, 9, 1, 7, 5, 6, 9, 8, 7, 4, 5, 1, 8, 6, 4, 4, 6, 9, 5, 7, 9, 9, 1, 6, 6, 8, 6, 9, 0, 3, 4, 8, 4, 2, 2, 5, 7, 2, 7, 3, 6, 5, 9, 2, 4, 6, 6, 7, 5, 9, 3, 2, 4, 9, 6, 6, 1, 3, 3, 3, 3, 6, 6, 8, 4, 1, 4, 3, 5, 8, 7, 7, 1, 6, 3, 7, 2, 0, 1, 9, 7, 4, 6, 3
OFFSET
0,1
COMMENTS
The odd infinite power tower function is h_o(x) = lim f(n,x) as n --> infinity, where f(n+1,x) = x^x^(f(n,x)) and f(1,x) = x. The even infinite power tower function h_e(x) is the same limit except with f(1,x) = x^x (see A194347). The limits exist if and only if 0 < x <= e^(1/e). If (1/e)^e <= x <= e^(1/e), then h_o(x) = h_e(x) = h(x) (the infinite power tower function-see the comments in A073230) and y = h(x) is a solution of x^y = y. If 0 < x < (1/e)^e, then h_o(x) < h_e(x), and two solutions of x^x^y = y are y = h_o(x) and y = h_e(x). For example, y = h_o(1/16) = 1/4 and y = h_e(1/16) = 1/2 are solutions of (1/16)^(1/16)^y = y.
h_o(1/17) and h_e(1/17) are irrational, and at least one of them is transcendental (see Sondow and Marques 2010).
REFERENCES
See the References in Sondow and Marques 2010.
LINKS
J. Sondow and D. Marques, Algebraic and transcendental solutions of some exponential equations, Annales Mathematicae et Informaticae 37 (2010) 151-164; see Definition 4.3, Figure 7, and top of p. 163.
EXAMPLE
0.204274736665518499175698745186446957991668690348422572736592466759324966133336...
MATHEMATICA
a = N[1/17, 100]; Do[a = (1/17)^(1/17)^a, {3000}]; RealDigits[a, 10, 100] // First
RealDigits[ Fold[ N[#2^#1, 128] &, 1/17, Table[1/17, {5710}]], 10, 105][[1]] (* Robert G. Wilson v, Mar 20 2012 *)
PROG
(PARI) solve(x=0, 1, 17^(-17^-x)-x) \\ Charles R Greathouse IV, Mar 20 2012
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Jonathan Sondow, Aug 27 2011
STATUS
approved