%I #28 Feb 08 2016 11:52:52

%S 1,2,4,10,12,32,36,572,1991,20857,27432,28763,155122,190271,288108,

%T 484709,1395499,9370521,12918396,22646948,49496125,73469408,172128260,

%U 645676547

%N Positions of the incrementally largest terms in continued fraction for 2^(1/3).

%C The corresponding records (or high-water marks) in A002945, the continued fraction for 2^(1/3), are {1, 3, 5, 8, 14, 15, 534, 7451, 12737, 22466, 68346, 148017, 217441, 320408, 533679, 4156269, 4886972, 10253793, ...} - see A268515.

%C It is not known if this sequence is infinite (i.e., whether the continued fraction expansion is bounded). [Davenport]. - _N. J. A. Sloane_, Feb 07 2016

%D H. Davenport, The Higher Arithmetic: An Introduction to the Theory of Numbers, Cambridge, 2008.

%t Use Max[ContinuedFraction[2^(1/3), n]] for some positive integer n, e.g. Max[ContinuedFraction[2^(1/3), 288108]].

%t cf = ContinuedFraction[2^(1/3), 20000000]; mx = 0; k = 1; lst = {}; While[k < 20000000, If[ cf[[k]] > mx, mx = cf[[k]]; AppendTo[lst, k]; Print[{k, cf[[k]]}]]; k++ ]; lst (* _Robert G. Wilson v_, Oct 24 2010 *)

%Y Cf. A002945, A268515.

%K nonn,more

%O 1,2

%A _John M. Campbell_, Oct 23 2010

%E a(19) from _Robert G. Wilson v_, Oct 24 2010

%E a(20)-a(21) from _Zak Seidov_, Feb 08 2016

%E a(22)-a(24) from _Hans Havermann_, Feb 08 2016