A206586 Numbers k such that the periodic part of the continued fraction of sqrt(k) has positive even length.

3, 6, 7, 8, 11, 12, 14, 15, 18, 19, 20, 21, 22, 23, 24, 27, 28, 30, 31, 32, 33, 34, 35, 38, 39, 40, 42, 43, 44, 45, 46, 47, 48, 51, 52, 54, 55, 56, 57, 59, 60, 62, 63, 66, 67, 68, 69, 70, 71, 72, 75, 76, 77, 78, 79, 80, 83, 84, 86, 87, 88, 90, 91, 92, 93, 94

Offset: 1

Author

T. D. Noe, Mar 19 2012

Keywords

• nonn

Comments

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.

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• P. J. Rippon and H. Taylor, Even and odd periods in continued fractions of square roots, Fibonacci Quarterly 42, May 2004, pp. 170-180.

Crossrefs

Cf. A003814 (period is odd), A206587.

Programs

Mathematica

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

PARI

cyc(cf) = {
  if(#cf==1, return(0)); \\ There is no cycle
  my(s=[]);
  for(k=2, #cf,
    s=concat(s, cf[k]);
    if(cf[k]==2*cf[1], return(s)) \\ Cycle found
  );
  0 \\ Cycle not found
}
select(n->(t=#cyc(contfrac(sqrt(n))))>0 && t%2==0, vector(100, n, n)) \\ Colin Barker, Oct 19 2014