A013644 Numbers k such that the continued fraction for sqrt(k) has period 4.

7, 14, 23, 28, 32, 33, 34, 47, 55, 60, 62, 75, 78, 79, 95, 96, 98, 119, 126, 128, 136, 138, 140, 141, 142, 155, 167, 174, 176, 180, 189, 192, 194, 215, 219, 220, 222, 223, 248, 252, 254, 266, 287, 299, 300, 305, 312, 315, 318, 320, 321, 322, 335, 359, 368, 377, 390, 392

Offset

1,1

References

Kenneth H. Rosen, Elementary Number Theory and Its Applications, Addison-Wesley, 1984, page 426 (but beware of errors!).

Links

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

Mack, Austin and Timothy Sawicki (2012) Pell's Equations Through Dynamical Systems [Broken link?]

Formula

see Mack, Austin and Timothy Sawicki(2012)

Example

The continued fraction for sqrt(7) is [2;1,1,1,4,...] with period 4, so 7 is in the sequence.  The continued fractions sqrt(3) = [1;1,2,...] with period 2 and sqrt(13) = [3;1,1,1,1,6,...] with period 5 do not have period 4, so 3 and 13 are not in the sequence. - Michael B. Porter, Sep 20 2016

Mathematica

cfp4Q[n_]:=Module[{sr=Sqrt[n]}, !IntegerQ[sr]&&Length[ ContinuedFraction[ sr][[2]]]==4]; Select[Range[500], cfp4Q] (* Harvey P. Dale, Jul 29 2014 *)

Crossrefs

Cf. A003285.

Sequence in context: A366112 A297426 A036556 * A178894 A356183 A050953

Adjacent sequences: A013641 A013642 A013643 * A013645 A013646 A013647

Keyword

nonn

Author

Clark Kimberling

Status

approved