A353621 Side c of primitive integer-sided triangles (a, b, c) whose angle B = 3*C.

8, 27, 64, 125, 125, 343, 343, 216, 343, 729, 343, 512, 729, 512, 1331, 729, 1728, 1331, 729, 1000, 1331, 2197, 1728, 1000, 1331, 2197, 1331, 1331, 2197, 3375, 2197, 4096, 3375, 1728, 2744, 2197, 2197, 4913, 4096, 2197, 2744, 4913, 3375, 6859, 2744, 4913, 4096, 6859, 3375, 4913, 8000

Offset

1,1

Comments

The triples (a, b, c) are displayed in increasing order of side b, and if sides b coincide then in increasing order of side c; hence, this sequence of sides c 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 opposite to the angle that is one-third of another one, for primitive integer-sided triangles.

All terms are cubes >= 8 (A000578). More generally, when c is the side of a primitive integer-sided triangles (a, b, c) whose angle B = m*C, then c = k^m, for some k >= 2.

Note that side c is never the largest side of the triangle.

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

Links

Table of n, a(n) for n=1..51.

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

Formula

a(n) = A353618(n, 3).

Example

According to inequalities between a, b, c, there exist 3 types of such triangles:
a < c < b with the middle side c = 8 of the first triple (3, 10, 8).
c < a < b with the smallest side c = 27 of the 2nd triple (35, 48, 27).
c < b < a with the smallest side c = 216 of the 8th triple (539, 510, 216), the first of this type.
The smallest side c for which there exist two distinct triangles with B = 3*C is for a(4) = a(5) = 125, and these sides c belong respectively to triples (112, 195, 125) and (279, 280, 125).

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(q^3); end if;
end do;
end do;

Crossrefs

Cf. A353618 (triples), A353619 (side a), A353620 (side b), this sequence (side c), A353622 (perimeter).

Cf. A343066 (similar, but with B = 2*C).

Cf. A000578.

Sequence in context: A062686 A093322 A017670 * A126200 A213491 A276919

Adjacent sequences: A353618 A353619 A353620 * A353622 A353623 A353624

Keyword

nonn

Author

Bernard Schott, May 07 2022

Status

approved