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%I #42 Nov 04 2023 02:53:43
%S 1,2,3,7,13,19,31,43,46,94,139,151,166,211,331,421,526,571,604,631,
%T 751,886,919,1291,1324,1366,1516,1621,1726,2011,2311,2566,2671,3004,
%U 3019,3334,3691,3931,4174,4846,5119,6211,6451,6679,6694,7606,8254,8779,8941,9739
%N Values of k at which the period of the continued fraction for sqrt(k) increases.
%C Periods of the fractions (sequence offset by one term) are given by A020640.
%C For n = 1 to 513 (the range of the b-file), the class number of the field Q(sqrt(a(n))) is 1 (computed with Mathematica). - _Emmanuel Vantieghem_, Mar 16 2017
%D Kenneth H. Rosen, Elementary Number Theory and Its Applications, Addison-Wesley, 1984, page 426 (but beware of errors!).
%H Patrick McKinley, <a href="/A013645/b013645.txt">Table of n, a(n) for n = 1..513</a> (first 200 terms from T. D. Noe)
%H H. C. Williams, <a href="https://doi.org/10.2307/2007664">A Numerical Investigation into the Length of the Period of the Continued Fraction Expansion of sqrt(D)</a>, Mathematics of Computation 36:154 (1981), 593-601 (see especially Tables 1 through 5 of this paper).
%e The continued fraction for sqrt(31) is {5; 1, 1, 3, 5, 3, 1, 1, 10}, the continued fraction for sqrt(43) is {6; 1, 1, 3, 1, 5, 1, 3, 1, 1, 12}, and there is no number between 31 and 43 whose square root produces a continued fraction whose period exceeds that of 31.
%t mx = -1; t = {}; Do[len = Length[ Last[ ContinuedFraction[ Sqrt[ n]]]]; If[len > mx, mx = len; AppendTo[t, n]], {n, 10^4}]; t
%Y Cf. A003285, A020640.
%K nonn,nice
%O 1,2
%A _Clark Kimberling_
%E More terms from _David W. Wilson_
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