%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