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%I #72 Aug 04 2018 06:42:25
%S 1,6,3,5,0,7,8,4,7,4,6,3,6,3,7,5,2,4,5,8,9,9,7,5,7,1,9,8,7,8,7,5,0,0,
%T 8,8,8,1,2,3,9,8,2,1,9,2,7,6,8,1,4,6,1,9,3,5,1,7,4,4,4,5,6,2,8,9,6,7,
%U 6,2,4,6,2,3,1,6,3,0,3,6,7,6,2,0,9,1,9,5,5,7,2,0,7,9,0,4,6,9,7,3,4,1,0,7
%N Decimal expansion of the solution to x^x^x = 3.
%C Let x(m) be the solution to the equation x^x^x^...^x = m, where x appears m times on the left hand side; e.g.,
%C decimal
%C m equation solution x(m) expansion
%C ==== ==================== ============= =============
%C 1 x = 1 1.00000000... A000007
%C 2 x^x = 2 1.55961046... A030798
%C 3 x^x^x = 3 1.63507847... this sequence
%C 4 x^x^x^x = 4 1.62036995...
%C 5 x^x^x^x^x = 5 1.59340881...
%C 6 x^x^x^x^x^x = 6 1.56864406...
%C 7 x^x^x^x^x^x^x = 7 1.54828598...
%C .
%C 10 x^x^x^x^...^x = 10 1.50849792...
%C .
%C 100 x^x^x^x^...^x = 100 1.44567285...
%C .
%C 1000 x^x^x^x^...^x = 1000 1.44467831...
%C .
%C Then x(1) < x(m) < x(3) for all m >= 4.
%C Let y(k/2) be the solution to the equation y^y^y^...^y = (k/2)*y^y, where y appears k times on the left hand side; e.g.,
%C decimal
%C k equation solution y(k/2) expansion
%C = ========================= =============== =========
%C 1 y = (1/2)*y^y 2 A000038
%C 2 y^y = (2/2)*y^y indeterminate
%C 3 y^y^y = (3/2)*y^y 1.6998419085...
%C 4 y^y^y^y = (4/2)*y^y 1.6396207046...
%C 5 y^y^y^y^y = (5/2)*y^y 1.5987769216...
%C 6 y^y^y^y^y^y = (6/2)*y^y 1.5694666408...
%C 7 y^y^y^y^y^y^y = (7/2)*y^y 1.5476452822...
%C .
%C What is lim_{k -> infinity} y(k/2)?
%C Lim_{m -> infinity} x(m) = e^(1/e). - _Jon E. Schoenfield_, Jul 23 2018
%C Lim_{k -> infinity} y(k/2) = e^(1/e). - _Jon E. Schoenfield_, Aug 01 2018
%e 1.635078474636375245899757198787500888...
%t RealDigits[ FindRoot[ x^x^x == 3, {x, 1}, WorkingPrecision -> 128][[1, 2]]][[1]] (* _Robert G. Wilson v_, Jun 13 2018 *)
%o (PARI) default(realprecision,333);
%o solve(x=1.6, 1.7, x^x^x-3) \\ _Joerg Arndt_, May 27 2018
%Y Cf. A000007, A000038, A030798.
%K nonn,cons
%O 1,2
%A _Juri-Stepan Gerasimov_, May 27 2018
%E More digits from _Michel Marcus_, _Joerg Arndt_, May 27 2018