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Second terms of continued fractions for power towers e, e^e, e^e^e, ...
4

%I #18 May 06 2013 04:26:43

%S 1,6,9,4

%N Second terms of continued fractions for power towers e, e^e, e^e^e, ...

%C It was conjectured (but remains unproved) that none of the power towers e, e^e, e^e^e, ... are integers. If so, the corresponding continued fractions contain at least 2 terms. If the conjecture fails, let the corresponding a(n) = 0.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/e.html">e.</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PowerTower.html">Power Tower</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Tetration#Open_questions">Tetration, Open questions</a>

%e a(3) = 9 because floor(1/frac(e^e^e)) = 9, since e^e^e ~ 3814279.10476.

%t $MaxExtraPrecision = Infinity; terms = 4; Map[Function[x, ContinuedFraction[x, 2][[2]]], NestList[Exp, E, terms - 1]]

%Y Cf. A003417, A064107, A159825, A225064, A004002.

%Y A056072 yields the first term of the continued fraction.

%K nonn,hard,more

%O 1,2

%A _Vladimir Reshetnikov_, Apr 25 2013