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A013644 Numbers n such that the continued fraction for sqrt(n) has period 4. 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 (list; graph; refs; listen; history; text; internal format)
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

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

Sequence in context: A246172 A297426 A036556 * A178894 A050953 A232825

Adjacent sequences:  A013641 A013642 A013643 * A013645 A013646 A013647

KEYWORD

nonn

AUTHOR

Clark Kimberling

STATUS

approved

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Last modified January 21 16:54 EST 2020. Contains 331114 sequences. (Running on oeis4.)