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A051537 Triangle read by rows: T(i,j) = lcm(i,j)/gcd(i,j). 9
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The first term of the k-th row is k. The first leading diagonal contains all 1's. The second leading diagonal contains twice triangular numbers = n*(n-1). a(p) = (p^3 - p^2 + 2)/2, where p is a prime. Proof: The p-th row is p, 2p, 3p, ..., (p-2)*p, (p-1)*p, 1 The sum = p*( 1+2+3+...+ (p-2) + (p-1)) + 1 = p*(p-1)*(p)/2 + 1 etc. - Robert G. Wilson v, May 10 2002

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

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

LINKS

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened

A. Jorza, Groups of Divisors: Solution to problem 10893, Amer. Math. Monthly, 2003, 441-443.

FORMULA

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

EXAMPLE

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; .....

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 *)

PROG

(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

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

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

CROSSREFS

Diagonals give A002378, A070260, A070261, A070262. Row sums give A056789.

Sequence in context: A217891 A322044 A010251 * A171999 A036038 A210237

Adjacent sequences:  A051534 A051535 A051536 * A051538 A051539 A051540

KEYWORD

nonn,tabl,changed

AUTHOR

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

EXTENSIONS

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

STATUS

approved

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Last modified June 19 05:26 EDT 2019. Contains 324218 sequences. (Running on oeis4.)