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 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. 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 (list; graph; refs; listen; history; text; internal format)
 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 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

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Last modified June 18 20:45 EDT 2021. Contains 345121 sequences. (Running on oeis4.)