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A242061
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Position within the triangular array A226314(n)/A054531(n) of rationals x/y such that x < y, gcd(x,y)=1, x+y odd and for the least y, {x, y} are integers such that x*y(y^2-x^2)/A006991(n) is a perfect square.
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0
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14, 2, 129, 52686, 29, 7, 9, 1274, 296125969, 12012350, 5, 1279281, 44, 302583265614, 780914, 90, 316, 2605, 106023820090609, 1754402265205275806, 7794, 72957466300254, 768323201, 40, 18505, 23, 6478321, 3966329, 326, 14280500082452241
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OFFSET
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1,1
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COMMENTS
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The triangle array A226314(n)/A054531(n) that enumerates all positive rationals x/y can be generalized to enumerate all ordered pairs {x, y} where x and y are natural numbers. For example, A243808 uses a subset of this triangular array to enumerate all primitive Pythagorean triples (PPT).
A006991(n) is the sequence of primitive congruent numbers and by definition there exists a PPT whose area is equal to k^2*A006991(n) for some integer k. a(n) is an enumeration of these PPT's and is a measure of the number of Pythagorean triangles that have to be searched to find a PPT with the least hypotenuse that has an area equal to k^2*A006991(n). If {x, y} are the generators of a PPT (a, b, c) where a = y^2-x^2, b = 2x*y, c=y^2+x^2 then its area = x*y(y^2-x^2). The Mathematica program limits searches to the first 12.5 million generated PPT's. All other results have been obtained from tables catalogued by Hisanori Mishima (see Links).
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LINKS
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EXAMPLE
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. -- --------------------------------------------------------
. 1: 1,1
. 2: 1,2 2,1
. 3: 1,3 2,3 3,1
. 4: 1,4 3,2 3,4 4,1
. 5: 1,5 2,5 3,5 4,5 5,1
. 6: 1,6 4,3 5,2 5,3 5,6 6,1
. 7: 1,7 2,7 3,7 4,7 5,7 6,7 7,1
. 8: 1,8 5,4 3,8 7,2 5,8 7,4 7,8 8,1
. 9: 1,9 2,9 7,3 4,9 5,9 8,3 7,9 8,9 9,1
. 10: 1,10 6,5 3,10 7,5 9,2 8,5 7,10 9,5 9,10 10,1
. 11: 1,11 2,11 3,11 4,11 5,11 6,11 7,11 8,11 9,11 10,11 11,1
. 12: 1,12 7,6 9,4 10,3 5,12 11,2 7,12 11,3 11,4 11,6 11,12 12,1 .
a(13)=44 and A006991(13)=34 so 34 is the 13th congruent number. a(13) gives the 44th term of the triangular array at index (8, 9). This gives (x, y) as (8, 9), it generates the PPT (17, 144, 145) and has an area 6^2*34 = 1224.
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MATHEMATICA
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lst1={5, 6, 7, 13, 14, 15, 21, 22, 23, 29, 30, 31, 34, 37, 38, 39, 41, 46, 47, 53, 55, 61, 62, 65, 69, 70, 71, 77, 78, 79, 85, 86, 87, 93, 94, 95, 101}; getpos[n0_] := (lst=0; Do[If[IntegerQ@Sqrt[m*n(m-n)(m+n)/n0]&&OddQ[m+n] && GCD[m, n]==1, (lst=m(m-1)/2+n; Break[])], {m, 2, 5000}, {n, 1, m-1}]; lst); SetAttributes[getpos, Listable]; getpos[lst1]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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