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Search: id:A051537
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| A051537 |
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Triangle T(i,j) read by rows: the j-th term of the i-th row is LCM(i,j)/GCD(i,j). |
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6
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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
(list; table; graph; listen)
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OFFSET
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1,2
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COMMENT
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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 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 (rgwv(AT)rgwv.com), 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 square-free number [A. Jorza]. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), May 03 2007
T(n,k) = A054531(n,k)*A164306(n,k). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Oct 30 2009]
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REFERENCES
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A. Jorza, Groups of Divisors, Solution to problem 10893, Amer. Math. Monthly, 2003, 441-443.
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EXAMPLE
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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; .....
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MATHEMATICA
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Flatten[ Table[ LCM[i, j] / GCD[i, j], {i, 1, 13}, {j, 1, i}]]
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CROSSREFS
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Diagonals give A002378, A070260, A070261, A070262. Row sums give A056789.
Sequence in context: A144867 A081520 A010251 this_sequence A036038 A078760 A103280
Adjacent sequences: A051534 A051535 A051536 this_sequence A051538 A051539 A051540
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KEYWORD
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nonn,tabl,new
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com) and Amarnath Murthy (amarnath_murthy(AT)yahoo.com), May 10, 2002
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EXTENSIONS
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More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), May 10 2002
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