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

Author

Clark Kimberling

Keywords

• nonn
• changed

Examples

			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
		

References

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
• Austin Mack and Timothy Sawicki, Pell's Equations Through Dynamical Systems, 2012. [Broken link?]

Crossrefs

Cf. A003285.

Programs

Mathematica

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

Formula

See Austin Mack and Timothy Sawicki(2012).