A353619 Side a of primitive integer-sided triangles (a, b, c) whose angle B = 3*C.

3, 35, 119, 112, 279, 20, 253, 539, 552, 91, 923, 533, 476, 1455, 224, 1504, 17, 799, 2159, 1513, 1476, 437, 1387, 3059, 2261, 1240, 3160, 4179, 2163, 748, 3212, 391, 1817, 5543, 3151, 4393, 5712, 1175, 2825, 7175, 5825, 2548, 5876, 189, 9099, 4077, 5859, 1736, 9352, 5768, 1189

Offset: 1

Author

Bernard Schott, May 07 2022

Keywords

• nonn

Comments

The triples (a, b, c) are displayed in increasing order of side b, and if sides b coincide then in increasing order of the side c; hence, this sequence of sides a is not increasing.

In the case B = 3*C, the corresponding metric relation between sides is c*a^2= (b-c)^2 * (b+c).

Equivalently, length of side common to the two angles, one being the triple of the other, of a primitive integer-sided triangle.

For the corresponding primitive triples and miscellaneous properties and references, see A353618.

Examples

			According to inequalities between a, b, c, there exist 3 types of such triangles:
a < c < b with the smallest side a = 3 of the first triple (3, 10, 8).
c < a < b with the middle side a = 35 of the 2nd triple (35, 48, 27).
c < b < a with the largest side a = 539 of the 8th triple (539, 510, 216), the first of this type.
		

Links

• The IMO Compendium, Problem 1, 46th Czech and Slovak Mathematical Olympiad 1997.

Crossrefs

Cf. A353618 (triples), this sequence (side a), A353620 (side b), A353621 (side c), A353622 (perimeter).

Cf. A343064 (similar, but with B = 2*C).

Programs

Maple

for b from 4 to 9000  do
  for q from 2 to floor((b-1)^(1/3)) do
a := (b-q^3) * sqrt(1+b/q^3);
if a= floor(a) and q^3 < b and igcd(a,b,q)=1 and (b-q^3) < a and a < b+q^3 then print(a); end if;
end do;
end do;

Formula

a(n) = A353618(n, 1).