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

1,2

Comments

This is A206586 and the squares, A000290.

Links

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

Mathematica

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

Prog

(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
(Python)
from itertools import count
from sympy.solvers.diophantine.diophantine import diop_DN
def A206587_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:n==1 or not len(diop_DN(n, -1)), count(max(startvalue, 1))) # Chai Wah Wu, Dec 21 2025

Crossrefs

Cf. A003814 (period has odd length).

Cf. A000290, A206586.

Sequence in context: A058654 A188435 A047517 * A192450 A039064 A187953

Adjacent sequences: A206584 A206585 A206586 * A206588 A206589 A206590

Keyword

nonn

Author

T. D. Noe, Mar 20 2012

Status

approved