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A020347
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Numbers k such that the continued fraction for sqrt(k) has period 6.
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4
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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
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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Robert Israel, Table of n, a(n) for n = 1..2500
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EXAMPLE
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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.
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MAPLE
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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
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MATHEMATICA
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Select[Range[500], Length[Last[ContinuedFraction[Sqrt[#]]]] == 6 &] (* Alonso del Arte, Mar 04 2018 *)
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CROSSREFS
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Cf. A157265.
Sequence in context: A042735 A156758 A089837 * A334103 A347343 A054864
Adjacent sequences: A020344 A020345 A020346 * A020348 A020349 A020350
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KEYWORD
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nonn
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AUTHOR
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David W. Wilson
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STATUS
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approved
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