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 A305187 Decimal expansion of the solution to x^x^x = 3. 0
 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, 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, 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 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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., decimal m equation solution x(m) expansion ==== ==================== ============= ============= 1 x = 1 1.00000000... A000007 2 x^x = 2 1.55961046... A030798 3 x^x^x = 3 1.63507847... this sequence 4 x^x^x^x = 4 1.62036995... 5 x^x^x^x^x = 5 1.59340881... 6 x^x^x^x^x^x = 6 1.56864406... 7 x^x^x^x^x^x^x = 7 1.54828598... . 10 x^x^x^x^...^x = 10 1.50849792... . 100 x^x^x^x^...^x = 100 1.44567285... . 1000 x^x^x^x^...^x = 1000 1.44467831... . Then x(1) < x(m) < x(3) for all m >= 4. 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., decimal k equation solution y(k/2) expansion = ========================= =============== ========= 1 y = (1/2)*y^y 2 A000038 2 y^y = (2/2)*y^y indeterminate 3 y^y^y = (3/2)*y^y 1.6998419085... 4 y^y^y^y = (4/2)*y^y 1.6396207046... 5 y^y^y^y^y = (5/2)*y^y 1.5987769216... 6 y^y^y^y^y^y = (6/2)*y^y 1.5694666408... 7 y^y^y^y^y^y^y = (7/2)*y^y 1.5476452822... . What is lim_{k -> infinity} y(k/2)? Lim_{m -> infinity} x(m) = e^(1/e). - Jon E. Schoenfield, Jul 23 2018 Lim_{k -> infinity} y(k/2) = e^(1/e). - Jon E. Schoenfield, Aug 01 2018 LINKS Table of n, a(n) for n=1..104. EXAMPLE 1.635078474636375245899757198787500888... MATHEMATICA RealDigits[ FindRoot[ x^x^x == 3, {x, 1}, WorkingPrecision -> 128][[1, 2]]][[1]] (* Robert G. Wilson v, Jun 13 2018 *) PROG (PARI) default(realprecision, 333); solve(x=1.6, 1.7, x^x^x-3) \\ Joerg Arndt, May 27 2018 CROSSREFS Cf. A000007, A000038, A030798. Sequence in context: A019165 A195490 A195471 * A065418 A228725 A286982 Adjacent sequences: A305184 A305185 A305186 * A305188 A305189 A305190 KEYWORD nonn,cons AUTHOR Juri-Stepan Gerasimov, May 27 2018 EXTENSIONS More digits from Michel Marcus, Joerg Arndt, May 27 2018 STATUS approved

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Last modified September 18 01:35 EDT 2024. Contains 375995 sequences. (Running on oeis4.)