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URL: https://oeis.org/A013654

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A013654
Positive nonsquare integers k such that each term of the regular continued fraction of sqrt(k) divides k.
1
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
OFFSET
1,1
COMMENTS
a(n) = n*(n+1) if n is odd and n*(n+2) if n is even. - Wesley Ivan Hurt, Nov 19 2013
From Rolf Knobel, Dec 27 2023: (Start)
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). -
Interleaving of A033996, A002939. Alternating mixture of A005563, A002378. (End)
REFERENCES
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!).
FORMULA
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
a(n) = n * (n + 1 + (1+(-1)^n)/2). - Wesley Ivan Hurt, Nov 19 2013
MAPLE
A013654:=n->n*(n+1+(1+(-1)^n)/2); seq(A013654(n), n=1..100); # Wesley Ivan Hurt, Nov 19 2013
MATHEMATICA
Table[n(n+1+(1+(-1)^n)/2), {n, 100}] (* Wesley Ivan Hurt, Nov 19 2013 *)
Table[If[OddQ[n], n(n+1), n(n+2)], {n, 50}] (* Harvey P. Dale, May 15 2021 *)
CROSSREFS
KEYWORD
nonn,easy
EXTENSIONS
More terms from David W. Wilson
Name clarified by Rolf Knobel, Dec 27 2023
STATUS
approved