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A335035
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Ordered perimeters of primitive integer triangles with two perpendicular medians.
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6
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54, 70, 104, 154, 170, 216, 252, 266, 352, 368, 418, 442, 464, 594, 598, 620, 638, 720, 740, 748, 792, 810, 902, 952, 962, 988, 1054, 1102, 1118, 1134, 1148, 1170, 1216, 1274, 1316, 1376, 1426, 1484, 1512, 1564, 1568, 1598, 1600, 1638, 1702, 1710, 1802, 1836, 1862
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OFFSET
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1,1
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COMMENTS
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The study of these integer triangles that have two perpendicular medians was proposed in the problem of Concours Général in 2007 in France (see link).
If medians drawn from A and B are perpendicular in centroid G, then a^2 + b^2 = 5 * c^2 (see Maths Challenge picture in link).
All terms are even because each triple is composed of one even side and two odd sides.
For the corresponding primitive triples and miscellaneous properties, see A335034.
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LINKS
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FORMULA
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EXAMPLE
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a(1) = 13 + 19 + 22 = 54 with 19^2 + 22^2 = 5 * 13^2 = 845.
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PROG
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(PARI) lista(nn) = {my(vm = List(), vt); for (u=1, nn, for (v=1, nn, if (gcd(u, v) == 1, vt = 0; if ((u/v > 3) && ((u-3*v) % 5), vt = [2*(u^2-u*v-v^2), u^2+4*u*v-v^2, u^2+v^2]); if ((u/v > 1) && (u/v < 2) && ((u-2*v) % 5), vt = [2*(u^2+u*v-v^2), -u^2+4*u*v+v^2, u^2+v^2]); if ((gcd(vt) == 1), listput(vm, vecsum(vt))); ); ); ); vecsort(vm); } \\ Michel Marcus, May 27 2020
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CROSSREFS
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Cf. A024364 (perimeters of primitive Pythagorean triangles).
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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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