A013645 Values of k at which the period of the continued fraction for sqrt(k) increases.

1, 2, 3, 7, 13, 19, 31, 43, 46, 94, 139, 151, 166, 211, 331, 421, 526, 571, 604, 631, 751, 886, 919, 1291, 1324, 1366, 1516, 1621, 1726, 2011, 2311, 2566, 2671, 3004, 3019, 3334, 3691, 3931, 4174, 4846, 5119, 6211, 6451, 6679, 6694, 7606, 8254, 8779, 8941, 9739

Offset

1,2

Comments

Periods of the fractions (sequence offset by one term) are given by A020640.

For n = 1 to 513 (the range of the b-file), the class number of the field Q(sqrt(a(n))) is 1 (computed with Mathematica). - Emmanuel Vantieghem, Mar 16 2017

References

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

Links

Patrick McKinley, Table of n, a(n) for n = 1..513 (first 200 terms from T. D. Noe)

H. C. Williams, A Numerical Investigation into the Length of the Period of the Continued Fraction Expansion of sqrt(D), Mathematics of Computation 36:154 (1981), 593-601 (see especially Tables 1 through 5 of this paper).

Example

The continued fraction for sqrt(31) is {5; 1, 1, 3, 5, 3, 1, 1, 10}, the continued fraction for sqrt(43) is {6; 1, 1, 3, 1, 5, 1, 3, 1, 1, 12}, and there is no number between 31 and 43 whose square root produces a continued fraction whose period exceeds that of 31.

Mathematica

mx = -1; t = {}; Do[len = Length[ Last[ ContinuedFraction[ Sqrt[ n]]]]; If[len > mx, mx = len; AppendTo[t, n]], {n, 10^4}]; t

Crossrefs

Cf. A003285, A020640.

Sequence in context: A210393 A045331 A053613 * A130272 A331948 A342529

Adjacent sequences: A013642 A013643 A013644 * A013646 A013647 A013648

Keyword

nonn,nice

Author

Clark Kimberling

Extensions

More terms from David W. Wilson

Status

approved