A020347 Numbers k such that the continued fraction for sqrt(k) has period 6.

19, 21, 22, 45, 52, 54, 57, 59, 70, 77, 88, 107, 111, 112, 114, 117, 131, 164, 165, 175, 178, 183, 187, 208, 216, 221, 232, 267, 270, 273, 275, 278, 280, 285, 294, 296, 303, 308, 350, 357, 371, 372, 374, 381, 387, 407, 418, 437, 456, 470, 498, 499, 507, 510, 514, 518

Offset

1,1

Comments

Includes A157265, corresponding to continued fractions [6*k+4,1,1,2,1,1,12*k+8,1,1,2,1,1,12*k+8,...]. - Robert Israel, Nov 21 2019

Links

Robert Israel, Table of n, a(n) for n = 1..2500

Example

The continued fraction for sqrt(19) is 4 + 1/(2 + 1/(1 + 1/(3 + 1/(1 + 1/(2 + 1/(8 + 1/(2 + 1/(1 + 1/(3 + 1/(1 + 1/(2 + 1/(8 + ..., which has period 6, so 19 is in the sequence.
The continued fraction for sqrt(20) is 4 + 1/(2 + 1/(8 + 1/(2 + 1/(8 + 1/(2 + 1/(8 + ..., which has a period of 2, so 20 is not in the sequence.

Maple

filter:= proc(n)
  not issqr(n) and nops(numtheory:-cfrac(sqrt(n), periodic, quotients)[2])=6
end proc:
select(filter, [$1..1000]); # Robert Israel, Nov 21 2019

Mathematica

Select[Range[500], Length[Last[ContinuedFraction[Sqrt[#]]]] == 6 &] (* Alonso del Arte, Mar 04 2018 *)

Crossrefs

Cf. A157265.

Sequence in context: A042735 A156758 A089837 * A334103 A347343 A054864

Adjacent sequences: A020344 A020345 A020346 * A020348 A020349 A020350

Keyword

nonn

Author

David W. Wilson

Status

approved