cp's OEIS Frontend

A335273 Even side of primitive triples for integer-sided triangles that have two perpendicular medians, the triples being ordered by increasing perimeter.

%I A335273 #24 Jan 14 2024 12:10:56
%S A335273 22,22,38,58,58,62,82,122,118,158,178,122,142,178,262,242,302,278,218,
%T A335273 202,358,362,262,382,418,298,478,398,398,302,482,562,502,362,458,638,
%U A335273 418,418,638,538,622,718,422,758,662,478,698,802,902,682,542,922,662,818
%N A335273 Even side of primitive triples for integer-sided triangles that have two perpendicular medians, the triples being ordered by increasing perimeter.
%C A335273 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).
%C A335273 For the corresponding primitive triples and miscellaneous properties, see A335034.
%C A335273 In each increasing triple (c,a,b), c is always the smallest odd side (A335036), but the even side a' can be either the middle side a (A335347) or the largest side b (A335348) (see formulas and examples for explanations).
%C A335273 The even side a' is not divisible by 4, and the odd prime factors of this even side term a' are all of the form 10*k +- 1.
%C A335273 The repetitions for 22, 58, 122,... correspond to even sides for triangles with distinct perimeters (see examples).
%C A335273 This sequence is not increasing: a(8) = 122 for triangle with perimeter = 266 and a(9) = 118 for triangle with perimeter = 352; hence the even side is not an increasing function of the perimeter of these triangles.
%H A335273 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 A335273 Maths Challenge, <a href="https://mathschallenge.net/view/perpendicular_medians">Perpendicular medians</a>, Problem with picture.
%H A335273 <a href="/index/O#Olympiads">Index to sequences related to Olympiads and other Mathematical competitions</a>.
%F A335273 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 A335273 --> 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 A335273 If 3 < u/v < 3+sqrt(10) then a' (even) < b' and the triple in increasing order is (c, a = a', b = b'),
%F A335273 if u/v > 3+sqrt(10) then a' (even) > b' and the triple in increasing order is (c, a = b', b = a').
%F A335273 --> 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 A335273 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 A335273 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 A335273 The triples (13, 19, 22), and (17, 22, 31) correspond to triangles with respective perimeters equal to 54 and 70, so a(1) = a(2) = 22.
%e A335273 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 A335273 -> For 1st class of triangles, u/v > 3:
%e A335273 (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 A335273 (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) = 178 = a' = b.
%e A335273 -> For 2nd class, 1 < u/v < 2:
%e A335273 (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) = 22 = a' = b.
%e A335273 (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) = 38 = a' = a.
%o A335273 (PARI)
%o A335273 mycmp(x, y) = {my(xp = vecsum(x), yp = vecsum(y)); if (xp!=yp, return (xp-yp)); return (x[1] - y[1]); }
%o A335273 triples(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, vt)); ); ); ); vecsort(apply(vecsort, Vec(vm)), mycmp); } \\ A335034
%o A335273 lista(nn) = my(w=triples(nn)); vector(#w, k, select(x->!(x%2), w[k])[1]); \\ _Michel Marcus_, Jun 10 2020
%Y A335273 Cf. A335034 (primitive triples), A335035 (corresponding perimeters), A335036 (smallest side), A335347 (middle side), A335348 (largest side), this sequence (even side), A335418 (largest odd side).
%K A335273 nonn
%O A335273 1,1
%A A335273 _Bernard Schott_, Jun 09 2020
%E A335273 More terms from _Michel Marcus_, Jun 10 2020