

A013645


Values of n at which period of continued fraction for sqrt(n) increases.


4



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
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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 bfile), 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, AddisonWesley, 1984, page 426 (but beware of errors!).


LINKS

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


EXAMPLE

The continued fraction for Sqrt(31) is 5, {1, 1, 3, 5, 3, 1, 1, 10} and 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 the period of which exceeds the one for 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 A172238
Adjacent sequences: A013642 A013643 A013644 * A013646 A013647 A013648


KEYWORD

nonn,nice


AUTHOR

Clark Kimberling


EXTENSIONS

More terms from David W. Wilson


STATUS

approved



