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A233593 Natural numbers n such that the continued fraction expansion sqrt(n) = c(1)+c(1)/(c(2)+c(2)/(c(3)+c(3)/....)) is aperiodic. 3

%I

%S 7,13,14,19,21,22,23,28,29,31,32,33,34,41,43,46,47,52,53,54,55,57,58,

%T 59,60,61,62,67,69,70,71,73,74,75,76,77,78,79,85,86,88,89,91,92,93,94,

%U 95,96,97,98,103,106,107

%N Natural numbers n such that the continued fraction expansion sqrt(n) = c(1)+c(1)/(c(2)+c(2)/(c(3)+c(3)/....)) is aperiodic.

%C For more details about this type of expansions, see A233582.

%C The cases with known periodic expansions, listed in A233592, all become periodic after just two leading terms. In contrast, the Blazys's expansion of sqrt(a(k)) for every member a(k) of this list remains a-periodic up to at least 1000 terms. It is therefore conjectured, though not proved, that these expansions are indeed aperiodic.

%H Stanislav Sykora, <a href="/A233593/b233593.txt">Table of n, a(n) for n = 1..200</a>

%H S. Sykora, <a href="http://dx.doi.org/10.3247/sl4math13.001">Blazys' Expansions and Continued Fractions</a>, Stans Library, Vol.IV, 2013, DOI 10.3247/sl4math13.001

%H S. Sykora, <a href="http://oeis.org/wiki/File:BlazysExpansions.txt">PARI/GP scripts for Blazys expansions and fractions</a>, OEIS Wiki

%e Blazys' expansion of sqrt(7), A233587, is {2,3,30,34,111,...}. Its first 1000 terms are all different. Hence, 7 is a member of this list.

%Y Cf. A233592.

%Y Cf. Blazys's expansions: A233582, A233584, A233585, A233586, A233587.

%K nonn

%O 1,1

%A _Stanislav Sykora_, Jan 06 2014

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Last modified January 22 05:11 EST 2019. Contains 319353 sequences. (Running on oeis4.)