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 A353620 Side b of primitive integer-sided triangles (a, b, c) whose angle B = 3*C. 3
 10, 48, 132, 195, 280, 357, 504, 510, 665, 792, 840, 840, 1035, 1288, 1485, 1575, 1740, 1848, 1872, 1890, 2233, 2496, 2604, 2610, 2640, 3003, 3069, 3520, 3536, 3885, 4095, 4368, 4560, 4620, 4662, 4680, 5291, 5712, 5904, 5928, 6006, 6579, 6765, 6992, 7462, 7480, 7568, 8037, 8385, 8415, 8820 (list; graph; refs; listen; history; text; internal format)
 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 the side c. 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 the triple of an other one, for primitive integer-sided triangle. Note that side b is never the smallest side of the triangle. For the corresponding primitive triples and miscellaneous properties and references, see A353618. LINKS The IMO Compendium, Problem 1, 46th Czech and Slovak Mathematical Olympiad 1997. FORMULA a(n) = A353618(n, 2). EXAMPLE According to inequalities between a, b, c, there exist 3 types of such triangles: a < c < b with the largest side b = 10 of the first triple (3, 10, 8). c < a < b with the largest side b = 48 of the 2nd triple (35, 48, 27). c < b < a with the middle side b = 510 of the 8th triple (539, 510, 216), the first of this type. The first side b for which there exist two distinct triangles with B = 3*C is for a(11) = a(12) = 840, and these sides b belong respectively to triples (923, 840, 343) and (533, 840, 512). 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(b); end if; end do; end do; CROSSREFS Cf. A353618 (triples), A353619 (side a), this sequence (side b), A353621 (side c), A353622 (perimeter). Cf. A343065 (similar, but with B = 2*C). Sequence in context: A121073 A210371 A195023 * A277229 A163724 A271638 Adjacent sequences:  A353617 A353618 A353619 * A353621 A353622 A353623 KEYWORD nonn AUTHOR Bernard Schott, May 07 2022 STATUS approved

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Last modified August 17 05:07 EDT 2022. Contains 356184 sequences. (Running on oeis4.)