A206587 Numbers k such that the periodic part of the continued fraction of sqrt(k) has even length.

1, 3, 4, 6, 7, 8, 9, 11, 12, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 30, 31, 32, 33, 34, 35, 36, 38, 39, 40, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 54, 55, 56, 57, 59, 60, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 75, 76, 77, 78, 79, 80, 81, 83, 84

Offset: 1

Author

T. D. Noe, Mar 20 2012

Keywords

• nonn

Comments

This is A206586 and the squares, A000290.

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A003814 (period has odd length).

Cf. A000290, A206586.

Programs

Mathematica

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

PARI

cyc(cf) = {
  if(#cf==1, return([])); \\ 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->#cyc(contfrac(sqrt(n)))%2==0, vector(400, n, n)) \\ Colin Barker, Oct 19 2014