A109042 -id:A109042 - cp's OEIS frontend

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

Mitch Harris, Jun 18 2005

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

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.

Cf. A000166, A109042, A152749 (partial sums).
Cf. A066043 (essentially the same), A000034 (=a(n)/n), A026741 (=a(n)/2).
Cf. A186421, A210530.

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

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

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

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

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

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

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

A051173 Triangle read by rows: T(n, k) = lcm(n, k).

Original entry on oeis.org

1, 2, 2, 3, 6, 3, 4, 4, 12, 4, 5, 10, 15, 20, 5, 6, 6, 6, 12, 30, 6, 7, 14, 21, 28, 35, 42, 7, 8, 8, 24, 8, 40, 24, 56, 8, 9, 18, 9, 36, 45, 18, 63, 72, 9, 10, 10, 30, 20, 10, 30, 70, 40, 90, 10, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 11, 12, 12, 12, 12, 60, 12, 84, 24, 36, 60, 132, 12

Offset: 1

Clark Kimberling

			Triangle begins (for the full array see A109042):
  [1]  1;
  [2]  2,  2;
  [3]  3,  6,  3;
  [4]  4,  4, 12,  4;
  [5]  5, 10, 15, 20,  5;
  [6]  6,  6,  6, 12, 30,  6;
  [7]  7, 14, 21, 28, 35, 42,  7;
  [8]  8,  8, 24,  8, 40, 24, 56,  8;

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened
Keith Bourque and Steve Ligh, Matrices associated with multiplicative functions, Linear Algebra and its Applications 216 (1995) 267-275.
Shaofang Hong and Raphael Loewy, Asymptotic behavior of eigenvalues of greatest common divisor matrices, Glasgow Mathematical Journal, Vol. 46, No. 3 (2004) 551-569.
Eric Weisstein's World of Mathematics, Least Common Multiple
Wikipedia, Least Common Multiple
Index entries for sequences related to lcm's

Cf. A109043 (column 2), A051193 (row sums), A000384 (central terms).

Cf. A109042 (variant), A003990, A002260, A075362, A051537, A050873.

a051173 = lcm
a051173_row n = a051173_tabl !! (n-1)
a051173_tabl = map (\x -> map (lcm x) [1..x]) [1..]
-- Reinhard Zumkeller, Aug 13 2013, Jul 07 2013

A051173 := proc(u,v) ilcm(u,v) ; end proc: # R. J. Mathar, Apr 07 2011

Table[LCM[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 30 2018 *)

T(n,k) = lcm(n,k);
tabl(nn) = for (n=1, nn, for (k=1, n, print1(T(n,k), ", ")); print;) \\ Michel Marcus, Jul 10 2017

T(n, 1) = T(n, n) = n. T(n, 2) = A109043(n). - R. J. Mathar, Apr 07 2011
T(n, k) = A075362(n, k)/A050873(n, k), 1 <= k <= n. - Reinhard Zumkeller, Apr 25 2011
T(n, k) = A051537(n, k) * A050873(n, k). - Reinhard Zumkeller, Jul 07 2013

A109044 a(n) = lcm(n,3).

Original entry on oeis.org

0, 3, 6, 3, 12, 15, 6, 21, 24, 9, 30, 33, 12, 39, 42, 15, 48, 51, 18, 57, 60, 21, 66, 69, 24, 75, 78, 27, 84, 87, 30, 93, 96, 33, 102, 105, 36, 111, 114, 39, 120, 123, 42, 129, 132, 45, 138, 141, 48, 147, 150, 51, 156, 159, 54, 165, 168, 57, 174, 177, 60, 183, 186, 63

Offset: 0

Mitch Harris, Jun 18 2005

			G.f. = 3*x + 6*x^2 + 3*x^3 + 12*x^4 + 15*x^5 + 6*x^6 + 21*x^7 + 24*x^8 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).
Index entries for sequences related to lcm's.

Cf. A051176, A099837, A109007 (gcd(n,3)), A109042.

[Lcm(n,3): n in [0..63]]; // Bruno Berselli, Mar 11 2011

A109044:=n->lcm(n,3): seq(A109044(n), n=0..100); # Wesley Ivan Hurt, Jul 24 2016

LCM[Range[0, 100], 3] (* Wesley Ivan Hurt, Jul 24 2016 *)

a(n)=lcm(n,3) \\ Charles R Greathouse IV, Oct 07 2015

[lcm(n,3)for n in range(0, 64)] # Zerinvary Lajos, Jun 07 2009

a(n) = 3*n/gcd(n,3) = 3*n/A109007(n).
From Bruno Berselli, Mar 11 2011: (Start)
G.f.: 3*x*(1+2*x+x^2+2*x^3+x^4)/(1-x^3)^2.
a(n) = 3*A051176(n);
a(n) = n*(7-2*A099837(n))/3 for n>0. (End)
From Wesley Ivan Hurt, Jul 24 2016: (Start)
a(n) = 2*a(n-3) - a(n-6) for n>5.
a(n) = 9*n/(5 + 4*cos(2*n*Pi/3)).
If n mod 3 = 0 then 3*floor(n/3), else 3*n. (End)
a(n) = n*(1 + 2*((n^2) mod 3)). - Timothy Hopper, Feb 23 2017
From Michael Somos, Mar 04 2017: (Start)
G.f.: 3 * x / (1 - x)^2 - 6 * x^3 / (1 - x^3)^2. -
a(n) = a(-n) for all n in Z. (End)
Sum_{k=1..n} a(k) ~ (7/6) * n^2. - Amiram Eldar, Nov 26 2022
Sum_{n>=1} (-1)^(n+1)/a(n) = 5*log(2)/9. - Amiram Eldar, Sep 08 2023

A109047 a(n) = lcm(n, 6).

Original entry on oeis.org

0, 6, 6, 6, 12, 30, 6, 42, 24, 18, 30, 66, 12, 78, 42, 30, 48, 102, 18, 114, 60, 42, 66, 138, 24, 150, 78, 54, 84, 174, 30, 186, 96, 66, 102, 210, 36, 222, 114, 78, 120, 246, 42, 258, 132, 90, 138, 282, 48, 294, 150, 102, 156, 318, 54, 330, 168, 114, 174, 354

Offset: 0

Mitch Harris, Jun 18 2005

Amiram Eldar, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,2,0,0,0,0,0,-1).
Index entries for sequences related to lcm's.

Cf. A060789, A089128, A109042.

[Lcm(n,6): n in [0..100]]; // Vincenzo Librandi, Apr 18 2011

LCM[Range[0,60],6] (* Harvey P. Dale, Nov 20 2022 *)

[lcm(n,6)for n in range(0, 60)] # Zerinvary Lajos, Jun 07 2009

a(n) = n*6/gcd(n, 6).
From R. J. Mathar, Apr 18 2011: (Start)
a(n) = 2*a(n-6) - a(n-12) = 6*A060789(n) = 6*n/A089128(n).
G.f.: 6*x*(1+x+x^2+2*x^3+5*x^4+x^5+5*x^6+2*x^7+x^8+x^9+x^10) / ( (x-1)^2*(1+x)^2*(1+x+x^2)^2*(x^2-x+1)^2 ). (End)
Sum_{k=1..n} a(k) ~ (7/4) * n^2. - Amiram Eldar, Nov 26 2022

A109049 a(n) = lcm(n, 8).

Original entry on oeis.org

0, 8, 8, 24, 8, 40, 24, 56, 8, 72, 40, 88, 24, 104, 56, 120, 16, 136, 72, 152, 40, 168, 88, 184, 24, 200, 104, 216, 56, 232, 120, 248, 32, 264, 136, 280, 72, 296, 152, 312, 40, 328, 168, 344, 88, 360, 184, 376, 48, 392, 200, 408, 104, 424, 216, 440, 56, 456, 232

Offset: 0

Mitch Harris, Jun 18 2005

Amiram Eldar, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,-1).
Index entries for sequences related to lcm's.

Cf. A106609, A109011, A109042.

[Lcm(n,8): n in [0..100]]; // Vincenzo Librandi, Apr 18 2011

LCM[Range[0,60],8] (* Harvey P. Dale, May 03 2019 *)

[lcm(n,8)for n in range(0, 59)] # Zerinvary Lajos, Jun 09 2009

a(n) = n*8/gcd(n, 8).
From R. J. Mathar, Apr 18 2011: (Start)
G.f.: 8*x*(x^2-x+1)*(x^12 + 2*x^11 + 4*x^10 + 3*x^9 + 4*x^8 + 4*x^7 + 7*x^6 + 4*x^5 + 4*x^4 + 3*x^3 + 4*x^2 + 2*x + 1)  / ( (x-1)^2 *(1+x)^2 *(x^2+1)^2 *(x^4+1)^2 ).
a(n) = 8*A106609(n) = 8*n/A109011(n). (End)
Sum_{k=1..n} a(k) ~ (43/16) * n^2. - Amiram Eldar, Nov 26 2022

A109046 a(n) = lcm(n, 5).

Original entry on oeis.org

0, 5, 10, 15, 20, 5, 30, 35, 40, 45, 10, 55, 60, 65, 70, 15, 80, 85, 90, 95, 20, 105, 110, 115, 120, 25, 130, 135, 140, 145, 30, 155, 160, 165, 170, 35, 180, 185, 190, 195, 40, 205, 210, 215, 220, 45, 230, 235, 240, 245, 50, 255, 260, 265, 270, 55, 280, 285, 290

Offset: 0

Mitch Harris, Jun 18 2005

Amiram Eldar, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,2,0,0,0,0,-1).
Index entries for sequences related to lcm's.

Cf. A060791, A109009, A109042.

[Lcm(n,5): n in [0..100]]; // Vincenzo Librandi, Apr 18 2011

a[n_] := LCM[n, 5]; Array[a, 60, 0] (* Amiram Eldar, Nov 26 2022 *)
LinearRecurrence[{0,0,0,0,2,0,0,0,0,-1},{0,5,10,15,20,5,30,35,40,45},60] (* Harvey P. Dale, Oct 08 2023 *)

[lcm(n,5)for n in range(0, 59)] # Zerinvary Lajos, Jun 07 2009

a(n) = n*5/gcd(n, 5) = 5*n/A109009(n) = 5*A060791(n).
G.f.: 5*x*(x^4+x^3+3*x^2+x+1)*(x^4+x^3-x^2+x+1) / ( (x-1)^2*(x^4+x^3+x^2+x+1)^2 ). - R. J. Mathar, Apr 18 2011
Sum_{k=1..n} a(k) ~ (21/10) * n^2. - Amiram Eldar, Nov 26 2022
Sum_{n>=1} (-1)^(n+1)/a(n) = 9*log(2)/25. - Amiram Eldar, Sep 08 2023

A109051 a(n) = lcm(n,10).

Original entry on oeis.org

0, 10, 10, 30, 20, 10, 30, 70, 40, 90, 10, 110, 60, 130, 70, 30, 80, 170, 90, 190, 20, 210, 110, 230, 120, 50, 130, 270, 140, 290, 30, 310, 160, 330, 170, 70, 180, 370, 190, 390, 40, 410, 210, 430, 220, 90, 230, 470, 240, 490, 50, 510, 260, 530, 270, 110, 280

Offset: 0

Mitch Harris, Jun 18 2005

Amiram Eldar, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,-1).
Index entries for sequences related to lcm's.

Cf. A106611, A109013, A109042.

[Lcm(n,10): n in [0..100]]; // Vincenzo Librandi, Apr 18 2011

q:= [seq(10/igcd(i,10),i=1..10)]:
[0,seq(seq((10*i+j)*q[j],j=1..10),i=0..10)]; # Robert Israel, Feb 23 2016

a[n_] := LCM[n, 10]; Array[a, 60, 0] (* Amiram Eldar, Nov 26 2022 *)

a(n) = lcm(n, 10); \\ Michel Marcus, Feb 23 2016

[lcm(n,10)for n in range(0, 57)] # Zerinvary Lajos, Jun 07 2009

a(n) = n*10/gcd(n, 10).
a(n) = 10*n/A109013(n) = 10*A106611(n). - R. J. Mathar, Apr 18 2011
G.f.: 10*x*(1 +x +3*x^2 +2*x^3 +x^4 +3*x^5 +7*x^6 +4*x^7 +9*x^8 +x^9 +9*x^10 +4*x^11 +7*x^12 +3*x^13 +x^14 +2*x^15 +3*x^16 +x^17 +x^18)/(1 -x^10)^2. - Robert Israel, Feb 23 2016
Sum_{k=1..n} a(k) ~ (63/20) * n^2. - Amiram Eldar, Nov 26 2022

A109053 a(n) = lcm(n,12).

Original entry on oeis.org

0, 12, 12, 12, 12, 60, 12, 84, 24, 36, 60, 132, 12, 156, 84, 60, 48, 204, 36, 228, 60, 84, 132, 276, 24, 300, 156, 108, 84, 348, 60, 372, 96, 132, 204, 420, 36, 444, 228, 156, 120, 492, 84, 516, 132, 180, 276, 564, 48, 588, 300, 204, 156, 636, 108, 660, 168

Offset: 0

Mitch Harris, Jun 18 2005

Muniru A Asiru, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,-1).
Index entries for sequences related to lcm's.

Cf. A051724, A109042.

List([0..60],n->Lcm(n,12)); # Muniru A Asiru, Mar 04 2019

[LCM(n, 12): n in [0..60]]; // G. C. Greubel, Mar 06 2019

Array[LCM[#,12]&,60,0] (* Harvey P. Dale, Mar 26 2015 *)

concat(0, Vec(12*x*(1 + x + x^2 + x^3 + 5*x^4 + x^5 + 7*x^6 + 2*x^7 + 3*x^8 + 5*x^9 + 11*x^10 + x^11 + 11*x^12 + 5*x^13 + 3*x^14 + 2*x^15 + 7*x^16 + x^17 + 5*x^18 + x^19 + x^20 + x^21 + x^22) / (1 - 2*x^12 + x^24) + O(x^40))) \\ Colin Barker, Mar 04 2019

for(n=0,60, print1(lcm(n,12), ", ")) \\ G. C. Greubel, Mar 06 2019

[lcm(n,12) for n in range(0,57)] # Zerinvary Lajos, Jun 09 2009

a(n) = n*12/gcd(n, 12).
a(n) = 12*A051724(n). - R. J. Mathar, Feb 12 2019
From Colin Barker, Mar 04 2019: (Start)
G.f.: 12*x*(1 + x + x^2 + x^3 + 5*x^4 + x^5 + 7*x^6 + 2*x^7 + 3*x^8 + 5*x^9 + 11*x^10 + x^11 + 11*x^12 + 5*x^13 + 3*x^14 + 2*x^15 + 7*x^16 + x^17 + 5*x^18 + x^19 + x^20 + x^21 + x^22) / (1 - 2*x^12 + x^24).
a(n) = 2*a(n-12) - a(n-24) for n>23.
(End)
Sum_{k=1..n} a(k) ~ (77/24) * n^2. - Amiram Eldar, Nov 26 2022

A109045 a(n) = lcm(n,4).

Original entry on oeis.org

0, 4, 4, 12, 4, 20, 12, 28, 8, 36, 20, 44, 12, 52, 28, 60, 16, 68, 36, 76, 20, 84, 44, 92, 24, 100, 52, 108, 28, 116, 60, 124, 32, 132, 68, 140, 36, 148, 76, 156, 40, 164, 84, 172, 44, 180, 92, 188, 48, 196, 100, 204, 52, 212, 108, 220, 56, 228, 116, 236, 60

Offset: 0

Mitch Harris, Jun 18 2005

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,2,0,0,0,-1).
Index entries for sequences related to lcm's.

Cf. A060819, A084351, A109008, A109042.

a109045 = lcm 4  -- Reinhard Zumkeller, Nov 25 2013

[Lcm(n,4): n in [0..100]]; // Vincenzo Librandi, Apr 18 2011

Table[LCM[n,4],{n,0,80}] (* Harvey P. Dale, May 02 2011 *)

a(n) = lcm(n, 4); \\ Michel Marcus, Mar 07 2018

a(n) = 4n/gcd(n, 4).
a(n) = A084351(n), n > 1. - R. J. Mathar, Aug 20 2008
From R. J. Mathar, Apr 18 2011: (Start)
G.f.: 4*x*(1+x+3*x^2+x^3+3*x^4+x^5+x^6) / ( (x-1)^2*(1+x)^2*(x^2+1)^2 ).
a(n) = 4*n/A109008(n) = 4*A060819(n). (End)
Sum_{k=1..n} a(k) ~ (11/8) * n^2. - Amiram Eldar, Nov 26 2022

A109048 a(n) = lcm(n, 7).

Original entry on oeis.org

0, 7, 14, 21, 28, 35, 42, 7, 56, 63, 70, 77, 84, 91, 14, 105, 112, 119, 126, 133, 140, 21, 154, 161, 168, 175, 182, 189, 28, 203, 210, 217, 224, 231, 238, 35, 252, 259, 266, 273, 280, 287, 42, 301, 308, 315, 322, 329, 336, 49, 350, 357, 364, 371, 378, 385, 56

Offset: 0

Mitch Harris, Jun 18 2005

Amiram Eldar, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,2,0,0,0,0,0,0,-1).
Index entries for sequences related to lcm's.

Cf. A106608, A109010, A109042.

[Lcm(n,7): n in [0..100]]; // Vincenzo Librandi, Apr 18 2011

LCM[Range[0, 100], 7] (* Wesley Ivan Hurt, Nov 17 2022 *)

[lcm(n,7)for n in range(0, 57)] # Zerinvary Lajos, Jun 09 2009

a(n) = n*7/gcd(n, 7).
a(n) = 7*A106608(n) = 7*n/A109010(n). - R. J. Mathar, Apr 18 2011
Sum_{k=1..n} a(k) ~ (43/14) * n^2. - Amiram Eldar, Nov 26 2022
Sum_{n>=1} (-1)^(n+1)/a(n) = 13*log(2)/49. - Amiram Eldar, Sep 08 2023

cp's OEIS Frontend

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

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A051173 Triangle read by rows: T(n, k) = lcm(n, k).

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A109044 a(n) = lcm(n,3).

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A109047 a(n) = lcm(n, 6).

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A109049 a(n) = lcm(n, 8).

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A109046 a(n) = lcm(n, 5).

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A109051 a(n) = lcm(n,10).

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