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A206586 Numbers n such that the periodic part of the continued fraction of sqrt(n) has positive even length. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

By making the length positive, we exclude squares. See A206587 for this sequence and the squares. All primes of the form 4k + 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.

MATHEMATICA

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

PROG

(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

CROSSREFS

Cf. A003814 (period has odd length).

Sequence in context: A047283 A155932 A298980 * A289176 A277851 A047557

Adjacent sequences:  A206583 A206584 A206585 * A206587 A206588 A206589

KEYWORD

nonn

AUTHOR

T. D. Noe, Mar 19 2012

STATUS

approved

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Last modified August 22 05:00 EDT 2019. Contains 326172 sequences. (Running on oeis4.)