A024354 -id:A024354 - cp's OEIS frontend

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

Clark Kimberling

nonn

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)

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

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

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).

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

Extended and corrected by David W. Wilson

A156678 Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, gcd (A, B) = 1, A < BA020884(n)).

Original entry on oeis.org

4, 12, 24, 15, 40, 60, 35, 84, 112, 63, 144, 180, 21, 99, 220, 264, 143, 312, 364, 45, 195, 420, 480, 255, 56, 544, 612, 77, 323, 684, 80, 760, 399, 840, 924, 117, 483, 1012, 1104, 55, 575, 1200, 140, 1300, 165, 675, 1404, 1512, 783, 176, 1624, 1740, 91, 221, 899

Offset: 1

Ant King, Feb 15 2009

The ordered sequence of A values is A020884(n) and the ordered sequence of B values is A020883(n) (allowing repetitions) and A024354(n) (excluding repetitions)

			As the first four primitive Pythagorean triples (ordered by increasing A) are (3,4,5), (5,12,13), (7,24,25) and (8,15,17), then a(1)=4, a(2)=12, a(3)=24 and a(4)=15.

Beiler, Albert H.: Recreations In The Theory Of Numbers, Chapter XIV, The Eternal Triangle, Dover Publications Inc., New York, 1964, pp. 104-134.
Sierpinski, W.; Pythagorean Triangles, Dover Publications, Inc., Mineola, New York, 2003.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Ron Knott, Right-angled Triangles and Pythagoras' Theorem

A020884, A020883, A024354, A156679, A042965, A156680, A156681, A042965.

Cf. A037213.

a156678 n = a156678_list !! (n-1)
a156678_list = f 1 1 where
   f u v | v > uu `div` 2        = f (u + 1) (u + 2)
         | gcd u v > 1 || w == 0 = f u (v + 2)
         | otherwise             = v : f u (v + 2)
         where uu = u ^ 2; w = a037213 (uu + v ^ 2)
-- Reinhard Zumkeller, Nov 09 2012

PrimitivePythagoreanTriplets[n_]:=Module[{t={{3,4,5}},i=4,j=5},While[i

a(n) = A020884(n) + A156680(n).

A137407 Numbers that cannot be the length of the long leg in any primitive Pythagorean triple.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 13, 14, 16, 17, 18, 19, 20, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 57, 58, 59, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 78, 79, 81, 82, 83, 85, 86, 87, 88

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 14 2008, Aug 28 2009

nonn

Complement of A024354. [R. J. Mathar, Sep 03 2009]

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A137409, A020883.

Definition reworded by R. J. Mathar, Sep 03 2009

A375463 Numbers appearing on all three positions in ordered primitive Pythagorean triples.

Original entry on oeis.org

221, 325, 377, 425, 493, 629, 697, 725, 925, 1025, 1073, 1189, 1325, 1517, 1537, 1769, 1885, 1961, 2173, 2257, 2405, 2501, 2665, 2701, 2993, 3145, 3233, 3293, 3445, 3485, 3649, 3869, 3965, 3977, 4453, 4505, 4717, 4745, 5141, 5185, 5353, 5429, 5777, 5785, 5917

Offset: 1

Piotr Lipski, Aug 16 2024

nonn

			221 is a term since the following primitive Pythagorean triples have 221 in first, second and third position: (221, 24420, 24421), (60, 221, 229), (21, 220, 221).

Mathematics Stack Exchange, Is there an integer that serves as the short leg of a primitive Pythagorean triple, the long leg of another, and the hypotenuse of a third?
Rémy Sigrist, PARI program

Intersection of A008846, A024352 and A024354.

Cf. A263728.

PARI
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a(n) == 1 (mod 4). - Hugo Pfoertner, Aug 18 2024

More terms from Rémy Sigrist, Aug 17 2024

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A020883 Ordered long legs of primitive Pythagorean triangles.

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A156678 Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, gcd (A, B) = 1, A < BA020884(n)).

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A137407 Numbers that cannot be the length of the long leg in any primitive Pythagorean triple.

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A375463 Numbers appearing on all three positions in ordered primitive Pythagorean triples.

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