A353619 - OEIS

%I #14 May 08 2022 15:15:45

%S 3,35,119,112,279,20,253,539,552,91,923,533,476,1455,224,1504,17,799,

%T 2159,1513,1476,437,1387,3059,2261,1240,3160,4179,2163,748,3212,391,

%U 1817,5543,3151,4393,5712,1175,2825,7175,5825,2548,5876,189,9099,4077,5859,1736,9352,5768,1189

%N Side a 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 the side c; hence, this sequence of sides a 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 common to the two angles, one being the triple of the other, of a primitive integer-sided 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, 1).

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

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

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

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

%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(a); end if;

%p end do;

%p end do;

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

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

%K nonn

%O 1,1

%A _Bernard Schott_, May 07 2022