A206586 - OEIS

%I #19 Jan 05 2025 19:51:39

%S 3,6,7,8,11,12,14,15,18,19,20,21,22,23,24,27,28,30,31,32,33,34,35,38,

%T 39,40,42,43,44,45,46,47,48,51,52,54,55,56,57,59,60,62,63,66,67,68,69,

%U 70,71,72,75,76,77,78,79,80,83,84,86,87,88,90,91,92,93,94

%N Numbers k such that the periodic part of the continued fraction of sqrt(k) has positive even length.

%C By making the length positive, we exclude squares. See A206587 for this sequence and the squares. All primes of the form 4m + 3 are here.

%H T. D. Noe, <a href="/A206586/b206586.txt">Table of n, a(n) for n = 1..1000</a>

%H P. J. Rippon and H. Taylor, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/42-2/quartrippon02_2004.pdf">Even and odd periods in continued fractions of square roots</a>, Fibonacci Quarterly 42, May 2004, pp. 170-180.

%t Select[Range[100], ! IntegerQ[Sqrt[#]] && EvenQ[Length[ContinuedFraction[Sqrt[#]][[2]]]] &]

%o (PARI)

%o cyc(cf) = {

%o   if(#cf==1, return(0)); \\ There is no cycle

%o   my(s=[]);

%o   for(k=2, #cf,

%o     s=concat(s, cf[k]);

%o     if(cf[k]==2*cf[1], return(s)) \\ Cycle found

%o   );

%o   0 \\ Cycle not found

%o }

%o select(n->(t=#cyc(contfrac(sqrt(n))))>0 && t%2==0, vector(100, n, n)) \\ _Colin Barker_, Oct 19 2014

%Y Cf. A003814 (period is odd), A206587.

%K nonn

%O 1,1

%A _T. D. Noe_, Mar 19 2012