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

Author

Amiram Eldar, Jul 01 2024

Keywords

• nonn

Examples

			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}.
		

Links

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

Crossrefs

Cf. A003285, A028832.

Programs

Mathematica

s[n_] := s[n] = If[IntegerQ@ Sqrt[n], 0, Union[ContinuedFraction[Sqrt[n]][[2]]]]; Select[Range[14000], s[#] == s[# + 1] &]