cp's OEIS Frontend

A335036 Smallest side c of the primitive triples (c,a,b) for integer triangles that have two perpendicular medians, ordered by increasing perimeter.

%I A335036 #26 Apr 09 2023 07:54:32
%S A335036 13,17,25,37,41,53,61,65,85,89,101,109,113,145,145,149,157,173,181,
%T A335036 185,193,197,221,229,233,241,257,265,269,281,277,289,293,313,317,337,
%U A335036 349,365,365,377,377,389,397,401,409,421,433,445,461
%N A335036 Smallest side c of the primitive triples (c,a,b) for integer triangles that have two perpendicular medians, ordered by increasing perimeter.
%C A335036 If medians drawn from A and B are perpendicular in centroid G, then a^2 + b^2 = 5 * c^2, hence c is always the smallest odd side (see link Maths Challenge).
%C A335036 c = u^2 + v^2 for some u and v (see formula), so this sequence is subsequence of A004431.
%C A335036 For the corresponding primitive triples and miscellaneous properties, see A335034.
%C A335036 The repetitions for 145, 365, 377,... correspond to smallest sides for triangles with distinct perimeters (see examples).
%C A335036 This sequence is not increasing a(30) = 281 for triangle with perimeter = 1134 and a(31) = 277 for triangle with perimeter = 1148. The smallest side is not an increasing function of the perimeter of these triangles.
%H A335036 Maths Challenge, <a href="https://mathschallenge.net/view/perpendicular_medians">Perpendicular medians</a>, Problem with picture.
%F A335036 a(n) = A335034(3n-2).
%F A335036 a(n) = A335035(n) - A335347(n) - A335348(n).
%F A335036 There exist two disjoint classes of such triangles, obtained with two distinct families of formulas: let u > v > 0 , u and v with different parities, gcd(u,v) = 1; if c is the smallest odd side, then:
%F A335036 1st class of triangles: c = u^2+v^2 with u/v > 3 and 5 doesn't divide u-3v,
%F A335036 2nd class of triangles: c = u^2+v^2 with 1 < u/v < 2 and 5 doesn't divide u-2v.
%e A335036 The triples (145, 178, 271) and (145, 191, 262) correspond to triangles with respective perimeters equal to 594 and 598, so a(14) = a(15) = 145.
%e A335036 The triples (365, 418, 701) and (365, 509, 638) correspond to triangles with respective perimeters equal to 1484 and 1512, so a(38) = a(39) = 365.
%o A335036 (PARI) mycmp(x, y) = {my(xp = vecsum(x), yp = vecsum(y)); if (xp!=yp, return (xp-yp)); return (x[1] - y[1]); }
%o A335036 lista(nn) = {my(vm = List(), vt, w); 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, vt));););); w = vecsort(apply(vecsort, Vec(vm)); , mycmp); vector(#w, k, w[k][1]);} \\ _Michel Marcus_, May 28 2020
%Y A335036 Subsequence of A004431.
%Y A335036 Cf. A335034 (primitive triples), A335035 (corresponding perimeters), A335347 (middle side), A335348 (largest side), A335273 (even side).
%K A335036 nonn
%O A335036 1,1
%A A335036 _Bernard Schott_, May 28 2020