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A374234
Number k such that the periods of the continued fractions of sqrt(k) and sqrt(k+1) have the same distinct terms.
1
7, 41, 44, 55, 74, 112, 135, 137, 207, 218, 275, 279, 314, 335, 389, 474, 611, 818, 874, 884, 986, 1007, 1009, 1129, 1313, 1325, 1462, 1465, 1824, 2330, 2831, 3201, 3502, 3575, 4927, 5520, 6204, 6623, 8150, 8945, 10989, 11627, 11834, 13033, 13727, 13775, 13888
OFFSET
1,1
LINKS
EXAMPLE
7 is a term since the period of the continued fraction of sqrt(7) is {1, 1, 1, 4} and the period of the continued fraction of sqrt(8) is {1, 4}. The set of distinct terms of both is {1, 4}.
44 is a term since the period of the continued fraction of sqrt(44) is {1, 1, 1, 2, 1, 1, 1, 12} and the period of the continued fraction of sqrt(45) is {1, 2, 2, 2, 1, 12}. The set of distinct terms of both is {1, 2, 12}.
MATHEMATICA
s[n_] := s[n] = If[IntegerQ@ Sqrt[n], 0, Union[ContinuedFraction[Sqrt[n]][[2]]]]; Select[Range[14000], s[#] == s[# + 1] &]
CROSSREFS
Sequence in context: A073501 A080666 A224718 * A272387 A338771 A161505
KEYWORD
nonn
AUTHOR
Amiram Eldar, Jul 01 2024
STATUS
approved