

A343891


List of primitive triples (a, b, c) for integersided 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.


7



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
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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 B337 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 64 < 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 a1 do
c := a*b/(2*ba);
if c=floor(c) and igcd(a, b, c)=1 and cb<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



