A013645 Values of k at which the period of the continued fraction for sqrt(k) sets a new record.

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

Author

Clark Kimberling

Keywords

• nonn
• nice

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

Examples

			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.
		

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).

Crossrefs

Cf. A003285, A020640.

Programs

Mathematica

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

Extensions

More terms from David W. Wilson