A020883 Ordered long legs of primitive Pythagorean triangles.

4, 12, 15, 21, 24, 35, 40, 45, 55, 56, 60, 63, 72, 77, 80, 84, 91, 99, 105, 112, 117, 120, 132, 140, 143, 144, 153, 156, 165, 168, 171, 176, 180, 187, 195, 208, 209, 220, 221, 224, 231, 240, 247, 252, 253, 255, 260, 264, 272, 273, 275, 285, 288, 299, 304, 308, 312, 323

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Comments

Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, (A, B) = 1, A < B); sequence gives values of B, sorted.

Any term in this sequence is given by f(m,n) = 2*m*n or g(m,n) = m^2 - n^2 where m and n are any two positive integers, m > 1, n < m, the greatest common divisor of m and n is 1, m and n are not both odd; e.g., f(m,n) = f(2,1) = 2*2*1 = 4. - Agola Kisira Odero, Apr 29 2016

All terms are composite. - Thomas Ordowski, Mar 12 2017

a(1) is the only power of 2. - Torlach Rush, Nov 08 2019

The first term appearing twice is 420 = a(75) = a(76) = A024410(1). - Giovanni Resta, Nov 11 2019

From Bernard Schott, May 05 2021: (Start)

Also, ordered sides a 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.

Example: a(2) = 12, because the second triple is (12, 10, 15) with side a = 12, satisfying 2/12 = 1/10 + 1/15 and 15-12 < 10 < 15+12.

The first term appearing twice 420 corresponds to triples (420, 310, 651) and (420, 406, 435), the second one is 572 = a(101) = a(102) = A024410(2) and corresponds to triples (572, 407, 962) and (572, 455, 770). The terms that appear more than once in this sequence are in A024410.

For the corresponding primitive triples and miscellaneous properties and references, see A343891. (End)

References

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

Links

• Giovanni Resta, Table of n, a(n) for n = 1..10000
• Ron Knott, Pythagorean Triples and Online Calculators

Crossrefs

Cf. A020882, A020884, A020885, A020886, A024354, A024410.

Triangles with 2/a = 1/b + 1/c: A343891 (triples), A020883 (side a), A343892 (side b), A343893 (side c), A343894 (perimeter).

Programs

Maple

for a from 4 to 325 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-bBernard Schott, May 05 2021

Extensions

Extended and corrected by David W. Wilson