

A020347


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


4



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



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



