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A051537
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Triangle read by rows: T(i,j) = lcm(i,j)/gcd(i,j) for 1 <= j <= i.
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11
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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
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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FORMULA
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EXAMPLE
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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;
...
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
(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
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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