A353621 - OEIS

%I #20 Apr 12 2023 21:40:05

%S 8,27,64,125,125,343,343,216,343,729,343,512,729,512,1331,729,1728,

%T 1331,729,1000,1331,2197,1728,1000,1331,2197,1331,1331,2197,3375,2197,

%U 4096,3375,1728,2744,2197,2197,4913,4096,2197,2744,4913,3375,6859,2744,4913,4096,6859,3375,4913,8000

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

%C 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.

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

%C Equivalently, length of side opposite to the angle that is one-third of another one, for primitive integer-sided triangles.

%C 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.

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

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

%H The IMO Compendium, <a href="https://imomath.com/othercomp/Czs/CzsMO97.pdf">Problem 1</a>, 46th Czech and Slovak Mathematical Olympiad 1997.

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

%e According to inequalities between a, b, c, there exist 3 types of such triangles:

%e a < c < b with the middle side c = 8 of the first triple (3, 10, 8).

%e c < a < b with the smallest side c = 27 of the 2nd triple (35, 48, 27).

%e c < b < a with the smallest side c = 216 of the 8th triple (539, 510, 216), the first of this type.

%e 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).

%p for b from 4 to 9000  do

%p   for q from 2 to floor((b-1)^(1/3)) do

%p a := (b-q^3) * sqrt(1+b/q^3);

%p 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;

%p end do;

%p end do;

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

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

%Y Cf. A000578.

%K nonn

%O 1,1

%A _Bernard Schott_, May 07 2022