A109043 a(n) = lcm(n,2).

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, 50, 26, 54, 28, 58, 30, 62, 32, 66, 34, 70, 36, 74, 38, 78, 40, 82, 42, 86, 44, 90, 46, 94, 48, 98, 50, 102, 52, 106, 54, 110, 56, 114, 58, 118, 60, 122, 62, 126, 64, 130, 66, 134

Offset: 0

Author

Mitch Harris, Jun 18 2005

Keywords

• nonn
• easy

Comments

Exponent of the dihedral group D(2n) = . - Arkadiusz Wesolowski, Sep 10 2013

Second column of table A210530. - Boris Putievskiy, Jan 29 2013

For n > 1, the basic period of A000166(k) (mod n) (Miska, 2016). - Amiram Eldar, Mar 03 2021

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Dorin Andrica, Sorin Rădulescu, and George Cătălin Ţurcaş, The Exponent of a Group: Properties, Computations and Applications, Disc. Math. and Applications, Springer, Cham (2020), 57-108.
• Piotr Miska, Arithmetic properties of the sequence of derangements, Journal of Number Theory, Vol. 163 (2016), pp. 114-145; arXiv preprint, arXiv:1508.01987 [math.NT], 2015. See p. 124 (p. 14 in the preprint).
• Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).
• Index entries for sequences related to lcm's.

Crossrefs

Cf. A000166, A109042, A152749 (partial sums).

Cf. A066043 (essentially the same), A000034 (=a(n)/n), A026741 (=a(n)/2).

Cf. A186421, A210530.

Programs

Haskell

a109043 = (lcm 2)
a109043_list = zipWith (*) [0..] a000034_list
-- Reinhard Zumkeller, Mar 31 2012

Magma

[0, 2, 2] cat [Exponent(DihedralGroup(n)) : n in [3..65]]; // Arkadiusz Wesolowski, Sep 10 2013

Mathematica

LCM[Range[0,70],2] (* Harvey P. Dale, Aug 19 2012 *)

PARI

a(n)=lcm(n,2) \\ Charles R Greathouse IV, Sep 24 2015

Python

def A109043(n): return n<<1 if n&1 else n # Chai Wah Wu, Aug 05 2024

Sage

[lcm(n,2) for n in range(0, 68)] # Zerinvary Lajos, Jun 07 2009

Formula

For n > 0: a(n) = A186421(n) + A186421(n+2).

a(n) = n*2 / gcd(n, 2).

a(n) = -(n*((-1)^n-3))/2. - Stephen Crowley, Feb 11 2007

From R. J. Mathar, Aug 20 2008: (Start)

a(n) = A066043(n), n > 1.

a(n) = 2*A026741(n).

G.f.: 2*x(1+x+x^2)/((1-x)^2*(1+x)^2). (End)

a(n) = n*A000034(n). - Paul Curtz, Mar 25 2011

E.g.f.: x*(2*cosh(x) + sinh(x)). - Stefano Spezia, May 09 2021

Sum_{k=1..n} a(k) ~ (3/4) * n^2. - Amiram Eldar, Nov 26 2022