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

13, 17, 25, 37, 41, 53, 61, 65, 85, 89, 101, 109, 113, 145, 145, 149, 157, 173, 181, 185, 193, 197, 221, 229, 233, 241, 257, 265, 269, 281, 277, 289, 293, 313, 317, 337, 349, 365, 365, 377, 377, 389, 397, 401, 409, 421, 433, 445, 461

Offset: 1

Author

Bernard Schott, May 28 2020

Keywords

• nonn

Comments

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 = u^2 + v^2 for some u and v (see formula), so this sequence is subsequence of A004431.

For the corresponding primitive triples and miscellaneous properties, see A335034.

The repetitions for 145, 365, 377,... correspond to smallest sides for triangles with distinct perimeters (see examples).

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.

Examples

			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.
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.
		

Links

• Maths Challenge, Perpendicular medians, Problem with picture.

Crossrefs

Subsequence of A004431.

Cf. A335034 (primitive triples), A335035 (corresponding perimeters), A335347 (middle side), A335348 (largest side), A335273 (even side).

Programs

PARI

mycmp(x, y) = {my(xp = vecsum(x), yp = vecsum(y)); if (xp!=yp, return (xp-yp)); return (x[1] - y[1]); }
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

Formula

a(n) = A335034(3n-2).

a(n) = A335035(n) - A335347(n) - A335348(n).

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:

1st class of triangles: c = u^2+v^2 with u/v > 3 and 5 doesn't divide u-3v,

2nd class of triangles: c = u^2+v^2 with 1 < u/v < 2 and 5 doesn't divide u-2v.