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A285579
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Circumdiameters of non-Pythagorean primitive Heronian triangles (no repetitions).
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2
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25, 65, 85, 125, 145, 169, 185, 205, 221, 265, 289, 305, 325, 365, 377, 425, 445, 481, 485, 493, 505, 533, 545, 565, 625, 629, 685, 689, 697, 725, 745, 785, 793, 841, 845, 865, 901, 905, 925, 949, 965, 985, 1025, 1037, 1105, 1145, 1157, 1165, 1189, 1205, 1241, 1261, 1285, 1313
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OFFSET
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1,1
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COMMENTS
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All Pythagorean triangles have their hypotenuse equal to their circumdiameter. This sequence excludes Pythagorean triangles from the superset of primitive Heronian triangles (PHT's) and lists all circumdiameters with integer values excluding repetitions. It appears that this sequence is a subset of A120961, from which first differs at a(45). Missing terms start 1073, 1537, 1961, 3277, 4181, 5713, … (see A329148). The table of PHT's used to generate this 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(1) = 25 as the Heronian triangle with sides (20, 15, 7) is primitive, is non-Pythagorean, has an area of 42 and has a circumdiameter of 25. It is the first occurrence of an integral circumdiameter for a non-Pythagorean PHT.
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MATHEMATICA
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lst1=ReadList["C:/primitive_heronian_triangles_1_10000.txt", {Number, Number, Number}]; lst = {}; Do[{a, b, c} = lst1[[n]]; s = (a + b + c)/2; A=Sqrt[s(s-a)(s-b)(s-c)]; If[IntegerQ[d=a*b*c/(2 A)]&&a!=d, AppendTo[lst, d]], {n, 1, Length[lst1]}]; Union@lst (* 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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