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%I #81 Nov 26 2022 04:03:42
%S 0,2,2,6,4,10,6,14,8,18,10,22,12,26,14,30,16,34,18,38,20,42,22,46,24,
%T 50,26,54,28,58,30,62,32,66,34,70,36,74,38,78,40,82,42,86,44,90,46,94,
%U 48,98,50,102,52,106,54,110,56,114,58,118,60,122,62,126,64,130,66,134
%N a(n) = lcm(n,2).
%C Exponent of the dihedral group D(2n) = <x, y | x^n = y^2 = 1, yxy = x^-1>. - _Arkadiusz Wesolowski_, Sep 10 2013
%C Second column of table A210530. - _Boris Putievskiy_, Jan 29 2013
%C For n > 1, the basic period of A000166(k) (mod n) (Miska, 2016). - _Amiram Eldar_, Mar 03 2021
%H Vincenzo Librandi, <a href="/A109043/b109043.txt">Table of n, a(n) for n = 0..10000</a>
%H Dorin Andrica, Sorin Rădulescu, and George Cătălin Ţurcaş, <a href="https://doi.org/10.1007/978-3-030-55857-4_4">The Exponent of a Group: Properties, Computations and Applications</a>, Disc. Math. and Applications, Springer, Cham (2020), 57-108.
%H Piotr Miska, <a href="https://doi.org/10.1016/j.jnt.2015.11.014">Arithmetic properties of the sequence of derangements</a>, Journal of Number Theory, Vol. 163 (2016), pp. 114-145; <a href="https://arxiv.org/abs/1508.01987">arXiv preprint</a>, arXiv:1508.01987 [math.NT], 2015. See p. 124 (p. 14 in the preprint).
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,-1).
%H <a href="/index/Lc#lcm">Index entries for sequences related to lcm's</a>.
%F For n > 0: a(n) = A186421(n) + A186421(n+2).
%F a(n) = n*2 / gcd(n, 2).
%F a(n) = -(n*((-1)^n-3))/2. - _Stephen Crowley_, Feb 11 2007
%F From _R. J. Mathar_, Aug 20 2008: (Start)
%F a(n) = A066043(n), n > 1.
%F a(n) = 2*A026741(n).
%F G.f.: 2*x(1+x+x^2)/((1-x)^2*(1+x)^2). (End)
%F a(n) = n*A000034(n). - _Paul Curtz_, Mar 25 2011
%F E.g.f.: x*(2*cosh(x) + sinh(x)). - _Stefano Spezia_, May 09 2021
%F Sum_{k=1..n} a(k) ~ (3/4) * n^2. - _Amiram Eldar_, Nov 26 2022
%t LCM[Range[0,70],2] (* _Harvey P. Dale_, Aug 19 2012 *)
%o (Sage) [lcm(n,2) for n in range(0, 68)] # _Zerinvary Lajos_, Jun 07 2009
%o (Haskell)
%o a109043 = (lcm 2)
%o a109043_list = zipWith (*) [0..] a000034_list
%o -- _Reinhard Zumkeller_, Mar 31 2012
%o (Magma) [0, 2, 2] cat [Exponent(DihedralGroup(n)) : n in [3..65]] // _Arkadiusz Wesolowski_, Sep 10 2013
%o (PARI) a(n)=lcm(n,2) \\ _Charles R Greathouse IV_, Sep 24 2015
%Y Cf. A000166, A109042, A152749 (partial sums).
%Y Cf. A066043 (essentially the same), A000034 (=a(n)/n), A026741 (=a(n)/2).
%Y Cf. A186421, A210530.
%K nonn,easy
%O 0,2
%A _Mitch Harris_, Jun 18 2005
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