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A013654
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Positive nonsquare integers k such that each term of the regular continued fraction of sqrt(k) divides k.
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1
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2, 8, 12, 24, 30, 48, 56, 80, 90, 120, 132, 168, 182, 224, 240, 288, 306, 360, 380, 440, 462, 528, 552, 624, 650, 728, 756, 840, 870, 960, 992, 1088, 1122, 1224, 1260, 1368, 1406, 1520, 1560, 1680, 1722, 1848, 1892, 2024, 2070, 2208, 2256, 2400, 2450, 2600
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OFFSET
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1,1
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COMMENTS
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a(n) = n*(n+1) if n is odd and n*(n+2) if n is even. - Wesley Ivan Hurt, Nov 19 2013
The first term of the continued fraction also divides k since it is half the last term of the periodic part. Except for a(1), the period is 2 (see A013642). -
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REFERENCES
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H. Davenport, The Higher Arithmetic. Cambridge Univ. Press, 7th edition, 1999, Table 1.
Kenneth H. Rosen, Elementary Number Theory and Its Applications, Addison-Wesley, 1984, page 426 (but beware of errors!).
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LINKS
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FORMULA
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a(2n) = 2*(n+1)*(2n), a(2n+1) = 2*(n+1)*(2n+1). - Frank Ellermann, Feb 22 2002
G.f.: (-2-6*x)/((x+1)^2*(x-1)^3). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 10 2009
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MAPLE
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MATHEMATICA
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Table[If[OddQ[n], n(n+1), n(n+2)], {n, 50}] (* Harvey P. Dale, May 15 2021 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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