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%I #35 Apr 09 2023 07:56:16
%S 54,70,104,154,170,216,252,266,352,368,418,442,464,594,598,620,638,
%T 720,740,748,792,810,902,952,962,988,1054,1102,1118,1134,1148,1170,
%U 1216,1274,1316,1376,1426,1484,1512,1564,1568,1598,1600,1638,1702,1710,1802,1836,1862
%N Ordered perimeters of primitive integer triangles with two perpendicular medians.
%C 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).
%C 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).
%C All terms are even because each triple is composed of one even side and two odd sides.
%C For the corresponding primitive triples and miscellaneous properties, see A335034.
%H Annales Concours Général, <a href="https://www.freemaths.fr/annales-composition-mathematiques-concours-general/concours-general-mathematiques-2007-sujet.pdf">Sujet Concours Général 2007</a>
%H Maths Challenge, <a href="https://mathschallenge.net/view/perpendicular_medians">Perpendicular medians</a>, Problem with picture.
%F a(n) = A335036(n) + A335347(n) + A335348(n).
%e a(1) = 13 + 19 + 22 = 54 with 19^2 + 22^2 = 5 * 13^2 = 845.
%o (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
%Y Cf. A024364 (perimeters of primitive Pythagorean triangles).
%Y Cf. A335034 (corresponding primitive triples), A335036 (smallest side), A335347 (middle side), A335348 (largest side), A335273 (even side).
%K nonn
%O 1,1
%A _Bernard Schott_, May 27 2020
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