cp's OEIS Frontend

A335347 Middle side of primitive triples for integer-sided triangles that have two perpendicular medians, the triples being ordered by increasing perimeter.

%I A335347 #24 Jun 05 2020 15:13:39
%S A335347 19,22,38,58,58,62,82,79,118,121,139,122,142,178,191,229,179,269,218,
%T A335347 202,241,251,262,341,311,298,319,398,398,302,389,319,421,362,458,401,
%U A335347 418,418,509,538,569,491,422,479,631,478,671,589,499
%N A335347 Middle side of primitive triples for integer-sided triangles that have two perpendicular medians, the triples being ordered by increasing perimeter.
%C A335347 If medians at A and B are perpendicular at the centroid G, then a^2 + b^2 = 5 * c^2 (see picture in Maths Challenge link), hence c is always the smallest side.
%C A335347 For the corresponding primitive triples and miscellaneous properties, see A335034; such a triangle with sides of integer lengths cannot be isosceles.
%C A335347 The middle side a with c < a < b is not divisible by 3, 4, or 5, and the odd prime factors of this middle side term a are all of the form 10*k +- 1.
%C A335347 In each increasing triple (c,a,b), c is the smallest odd side (A335036), but the middle side a can be either the even side (A335273) or the largest odd side (see formulas and examples for explanations).
%C A335347 The consecutive repetitions for 58, 398, 418,... correspond to middle sides for triangles with distinct perimeters (see examples).
%C A335347 This sequence is not increasing: a(7) = 82 for triangle with perimeter = 252 and a(8) = 79 for triangle with perimeter = 266; hence the middle side is not an increasing function of the perimeter of these triangles.
%H A335347 Maths Challenge, <a href="https://mathschallenge.net/view/perpendicular_medians">Perpendicular medians</a>, Problem with picture.
%F A335347 a(n) = A335034(3n-1).
%F A335347 a(n) = A335035(n) - A335036(n) - A335348(n).
%F A335347 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; a' is the even side and b' the largest odd side.
%F A335347 --> 1st class of triangles: (a',b',c) = (2*(u^2-uv-v^2), u^2+4*u*v-v^2, u^2+v^2) with u/v > 3 and 5 doesn't divide u-3v.
%F A335347 If 3 < u/v < 3+sqrt(10) then a' (even) < b' and the triple in increasing order is (c, a = a', b = b'),
%F A335347 If u/v > 3+sqrt(10) then a' (even) > b' and the triple in increasing order is (c, a = b', b = a').
%F A335347 --> 2nd class of triangles: (a',b',c) = (2*(u^2+uv-v^2), -u^2+4*u*v+v^2, u^2+v^2) with 1 < u/v < 2 and 5 doesn't divide u-2v.
%F A335347 If 1 < u/v < (1+sqrt(10))/3 then a' (even) < b' and the triple in increasing order  is (c, a = a', b = b'),
%F A335347 If (1+sqrt(10))/3 < u/v < 2 then a' (even) > b' and the triple in increasing order is (c, a = b', b = a').
%e A335347 The triples (37, 58, 59) and (41, 58, 71) correspond to triangles with respective perimeters equal to 154 and 170, so a(4) = a(5) = 58.
%e A335347 --> For 1st class of triangles, u/v > 3:
%e A335347 (u,v) = (4,1), then 3 < u/v < 3+sqrt(10) and  (c,a,b) = (c,a',b') = (17,22,31); the relation is 22^2 + 31^2 = 5 * 17^2 = 1445 with a(2) = 22 = a = a',
%e A335347 (u,v) = (10,1), then u/v > 3+sqrt(10) and (c,a,b) = (c,b',a') = (101, 139, 178), the relation is 139^2 + 178^2 = 5 * 101^2 = 51005 with a(11) = 139 = a = b'.
%e A335347 --> For 2nd class, 1 < u/v < 2:
%e A335347 (u,v) = (3,2), then (1+sqrt(10))/3 < u/v < 2 and (c,a,b) = (c,b',a') = (13,19,22), the relation is 19^2 + 22^2 = 5 * 13^2 = 845 with a(1) = 19 = a = b'
%e A335347 (u,v) = (4,3), then 1 < u/v < (1+sqrt(10))/3  and (c,a,b) = (c,a',b') = (25,38,41); the relation is 38^2 + 41^2 = 5 * 25^2 = 3125 with a(3) = 22 = a = a'.
%o A335347 (PARI) mycmp(x, y) = {my(xp = vecsum(x), yp = vecsum(y)); if (xp!=yp, return (xp-yp)); return (x[1] - y[1]); }
%o A335347 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][2]); }  \\ _Michel Marcus_, Jun 03 2020
%Y A335347 Cf. A335034 (primitive triples), A335035 (corresponding perimeters), A335036 (smallest side), A335348 (largest side), A335273 (even side).
%K A335347 nonn
%O A335347 1,1
%A A335347 _Bernard Schott_, Jun 03 2020