A374234 Number k such that the periods of the continued fractions of sqrt(k) and sqrt(k+1) have the same distinct terms.

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

Amiram Eldar, Table of n, a(n) for n = 1..216

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

Cf. A003285, A028832.

Sequence in context: A073501 A080666 A224718 * A272387 A338771 A161505

Adjacent sequences: A374231 A374232 A374233 * A374235 A374236 A374237

Keyword

nonn

Author

Amiram Eldar, Jul 01 2024

Status

approved