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A013646
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Least m such that continued fraction for sqrt(m) has period n.
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14
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1, 2, 3, 41, 7, 13, 19, 58, 31, 106, 43, 61, 46, 193, 134, 109, 94, 157, 139, 337, 151, 181, 166, 586, 271, 457, 211, 949, 334, 821, 379, 601, 463, 613, 331, 1061, 478, 421, 619, 541, 526, 1117, 571, 1153, 604, 1249, 694, 1069, 631, 1021, 1051, 1201, 751, 1669, 886
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OFFSET
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0,2
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COMMENTS
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In a search of fractions up to sqrt(1650241399), the smallest length not yet seen is 97921. The next unseen lengths are 101679, 102181 and 102407. After 145 more missing odd lengths, the first even length not seen is 107292. This would suggest that A215485 may be exclusively odd after an early 2, but beware the law of small numbers! - Patrick McKinley, Aug 24 2012
a(97921) = 1664155249, a(101679) = 1654486681, a(102181) = 1682919001, a(102407) = 1680133849, a(107292) = 1651931884, thus 107292 is not in A215485. - Chai Wah Wu, Jun 08 2017
a(999213) = 133511789629, a(1000000) = 98814608764. - Michael Hortmann, Mar 20 2023
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REFERENCES
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Kenneth H. Rosen, Elementary Number Theory and Its Applications, Addison-Wesley, 1984, page 426 (but beware of errors!).
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LINKS
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MATHEMATICA
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a[n_] := Catch[For[m = 1, True, m++, If[Length[ Last[ ContinuedFraction[ Sqrt[m] ]]] == n, Print[m]; Throw[m] ]]]; Table[a[n], {n, 0, 54}](* Jean-François Alcover, May 15 2012 *)
Flatten[Table[Position[Table[{s=Sqrt[n]}; If[IntegerQ[s], 0, Length[ ContinuedFraction[s] [[2]]]], {n, 2000}], i, {1}, 1], {i, 0, 60}]] (* Harvey P. Dale, Sep 15 2013 *)
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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STATUS
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approved
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