A051537 Triangle read by rows: T(i,j) = lcm(i,j)/gcd(i,j) for 1 <= j <= i.

1, 2, 1, 3, 6, 1, 4, 2, 12, 1, 5, 10, 15, 20, 1, 6, 3, 2, 6, 30, 1, 7, 14, 21, 28, 35, 42, 1, 8, 4, 24, 2, 40, 12, 56, 1, 9, 18, 3, 36, 45, 6, 63, 72, 1, 10, 5, 30, 10, 2, 15, 70, 20, 90, 1, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 1, 12, 6, 4, 3, 60, 2, 84, 6, 12, 30, 132, 1, 13, 26, 39

Offset: 1

Author

N. J. A. Sloane and Amarnath Murthy, May 10 2002

Keywords

• nonn
• tabl

Comments

From Robert G. Wilson v, May 10 2002: (Start)

The first term of the k-th row is k. The first leading diagonal contains all 1's. The second leading diagonal contains twice the triangular numbers = n*(n-1).

For p prime, the sum of the p-th row is (p^3 - p^2 + 2)/2.

Proof: The p-th row is p, 2*p, 3*p, ..., (p-2)*p, (p-1)*p, 1. The sum of the row = p*(1 + 2 + 3 + ... + (p-2) + (p-1)) + 1 = p*(p-1)*p/2 + 1 = (p^3 - p^2 + 2)/2. (End) [Edited by Petros Hadjicostas, May 27 2020]

In the square array where T(i,j) = T(j,i), the natural extension of the triangle, each set of rows and columns with common indices [d1, d2, ..., ds] define a group multiplication table on their grid, if the d1, d2, ..., ds are the set of divisors of a squarefree number [A. Jorza]. - R. J. Mathar, May 03 2007

T(n,k) is the minimum number of squares necessary to fill a rectangle with sides of length n and k. - Stefano Spezia, Oct 06 2018

Examples

			Triangle T(n,k) (with rows n >= 1 and columns k = 1..n) begins
  1;
  2,  1;
  3,  6,  1;
  4,  2, 12,  1;
  5, 10, 15, 20,  1;
  6,  3,  2,  6, 30,  1;
  7, 14, 21, 28, 35, 42,  1;
  8,  4, 24,  2, 40, 12, 56,  1;
  ...
		

Links

• Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened
• Andrei Jorza, Groups of Divisors: Solution to problem 10893, Amer. Math. Monthly, 2003, 441-443.

Crossrefs

Diagonals give A002378, A070260, A070261, A070262.

Row sums give A056789.

Programs

GAP

Flat(List([1..13],n->List([1..n],k->Lcm(n,k)/Gcd(n,k)))); # Muniru A Asiru, Oct 06 2018

Haskell

a051537 n k = a051537_tabl !! (n-1) !! (k-1)
a051537_row n = a051537_tabl !! (n-1)
a051537_tabl = zipWith (zipWith div) a051173_tabl a050873_tabl
-- Reinhard Zumkeller, Jul 07 2013

Magma

/* As triangle */ [[Lcm(n,k)/Gcd(n,k): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Oct 07 2018

Maple

T:=proc(n,k) n*k/gcd(n,k)^2; end proc: seq(seq(T(n,k),k=1..n),n=1..13); # Muniru A Asiru, Oct 06 2018

Mathematica

Flatten[ Table[ LCM[i, j] / GCD[i, j], {i, 1, 13}, {j, 1, i}]]
T[n_,k_]:=n*k/GCD[n,k]^2; Flatten[Table[T[n,k],{k,1,13},{n,1,k}]] (* Stefano Spezia, Oct 06 2018 *)

Formula

T(n,k) = A054531(n,k)*A164306(n,k). - Reinhard Zumkeller, Oct 30 2009

T(n,k) = A051173(n,k) / A050873(n,k). - Reinhard Zumkeller, Jul 07 2013

T(n,k) = n*k/gcd(n,k)^2. - Stefano Spezia, Oct 06 2018

Extensions

More terms from Robert G. Wilson v, May 10 2002