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A277632
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The ordered integer image of the 1-to-1 mapping of primitive Heronian triples (PHT) into the integers using Cantor's pairing function for triples (N^3 -> N).
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1
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1381, 2931, 5156, 58658, 70135, 79012, 89680, 106966, 152084, 171416, 197522, 212885, 266098, 295306, 400078, 434790, 675720, 789403, 863969, 866606, 917338, 936413, 1085618, 1149892, 1242687, 1432297, 1628115, 2116668, 2241911, 2250397, 2418925, 2694694, 2699343, 3022126, 3036895, 3059130
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OFFSET
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1,1
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COMMENTS
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This mapping of the Heronian triple (a,b,c) to an integer is unique and uses Cantor's pairing function K(i,j) = (i+j)(i+j+1)/2+j so that (a,b,c) -> K(K(a,b),c). The table of PHT's used to generate the sequence was obtained from lists generated by Sascha Kurz (see Link). The list contains a triple for every possible PHT with a maximum side length of 10000. The triples are in the form (a,b,c) where a >= b >= c and where a <= 10000.
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LINKS
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EXAMPLE
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A PHT with sides (a,b,c) = (21,20,13) maps to K(K(21,20),13) = K(881,13) = 400078 = a(15), where Cantor's pairing function K is simply A001477 in its two-argument tabular form A001477(k, n) = n + (k+n)*(k+n+1)/2.
A PHT with sides (a,b,c) = (29,21,20) maps to K(K(29,21),20) = 866606 = a(20). This is a primitive Pythagorean triangle (thus also a primitive Heronian triangle), listed as term a(5)=33 in A277557.
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MATHEMATICA
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Cantor[i_, j_] := (i+j)(i+j+1)/2+j; nn=50; lst1=ReadList["C:/primitive_heronian_triangles_1_10000.txt", {Number, Number, Number}]; lst2=Select[lst1, #[[1]]<=2 nn &]; lst={}; Do[({a, b, c}=lst2[[n]]; k=Cantor[Cantor[a, b], c]; AppendTo[lst, k]), {n, 1, Length[lst2]}]; Sort[Select[lst, #<Cantor[Cantor[nn, nn], nn] &]] (* For download of file of primitive Heronian triples see Link *)
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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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