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 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 STATUS approved

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Last modified March 30 07:38 EDT 2023. Contains 361606 sequences. (Running on oeis4.)