A343891 List of primitive triples (a, b, c) for integer-sided triangles where side a is the harmonic mean of the 2 other sides b and c, i.e., 2/a = 1/b + 1/c with b < a < c.

4, 3, 6, 12, 10, 15, 15, 12, 20, 21, 15, 35, 24, 21, 28, 35, 30, 42, 40, 36, 45, 45, 35, 63, 55, 40, 88, 56, 44, 77, 60, 55, 66, 63, 56, 72, 72, 52, 117, 77, 63, 99, 80, 65, 104, 84, 78, 91, 91, 70, 130, 99, 90, 110, 105, 77, 165, 112, 105, 120, 117, 99, 143, 120, 85, 204, 132, 102, 187

Offset

1,1

Comments

The triples (a, b, c) are displayed in increasing order of side a, and if sides a coincide then in increasing order of the side b.

When sides satisfy 2/a = 1/b + 1/c, or a = 2*b*c/(b+c) then a is always the middle side with b < a < c.

Equivalent relations: the heights and sines satisfy 2*h_a = h_b + h_c and 2/sin(A) = 1/sin(B) + 1/sin(C).

Inequalities between sides: a/2 < b < a < c < b*(1+sqrt(2)).

References

V. Lespinard & R. Pernet, Trigonométrie, Classe de Mathématiques élémentaires, programme 1962, problème B-337 p. 179, André Desvigne.

Links

Table of n, a(n) for n=1..69.

Example

(4, 3, 6) is the first triple with 2/4 = 1/3 + 1/6 and 6-4 < 3 < 6+4.
The table begins:
   4,  3,  6;
  12, 10, 15;
  15, 12, 20;
  21, 15, 35;
  24, 21, 28;
  35, 30, 42;
  ...

Maple

for a from 4 to 200 do
for b from floor(a/2)+1 to a-1 do
c := a*b/(2*b-a);
if c=floor(c) and igcd(a, b, c)=1 and c-b<a then print(a, b, c); end if;
end do;
end do;

Crossrefs

Cf. A020883 (side a), A343892 (side b), A343893 (side c), A343894 (perimeter).

Sequence in context: A276202 A215336 A328650 * A232328 A276229 A077955

Adjacent sequences: A343888 A343889 A343890 * A343892 A343893 A343894

Keyword

nonn,tabf

Author

Bernard Schott, May 03 2021

Status

approved