A020884 Ordered short legs of primitive Pythagorean triangles.

3, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 20, 21, 23, 24, 25, 27, 28, 28, 29, 31, 32, 33, 33, 35, 36, 36, 37, 39, 39, 40, 41, 43, 44, 44, 45, 47, 48, 48, 49, 51, 51, 52, 52, 53, 55, 56, 57, 57, 59, 60, 60, 60, 61, 63, 64, 65, 65, 67, 68, 68, 69, 69, 71, 72, 73, 75, 75, 76, 76, 77

Offset

1,1

Comments

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

Union of A081874 and A081925. - Lekraj Beedassy, Jul 28 2006

Each term in this sequence is given by f(m,n) = m^2 - n^2 or g(m,n) = 2mn where m and n are relatively prime positive integers with m > n, m and n not both odd. For example, a(1) = f(2,1) = 2^2 - 1^2 = 3 and a(4) = g(4,1) = 2*4*1 = 8. - Agola Kisira Odero, Apr 29 2016

All powers of 2 greater than 4 (2^2) are terms, and are generated by the function g(m,n) = 2mn. - Torlach Rush, Nov 08 2019

Links

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Reinhard Zumkeller)

P. Alfeld, Pythagorean Triples (broken link)

Nick Exner, Generating Pythagorean Triples. This was originally a Java applet (1998), modified by Michael McKelvey in 2001 and redone as an HTML page with JavaScript by Evan Ramos in 2014.

W. A. Kehowski, Pythagorean Triples.

Ron Knott, Pythagorean Triples and Online Calculators

Mathematica

shortLegs = {}; amx = 99; Do[For[b = a + 1, b < (a^2/2), c = (a^2 + b^2)^(1/2); If[c == IntegerPart[c] && GCD[a, b, c] == 1, AppendTo[shortLegs, a]]; b = b + 2], {a, 3, amx}]; shortLegs (* Vladimir Joseph Stephan Orlovsky, Aug 07 2008 *)

Prog

(Haskell)
a020884 n = a020884_list !! (n-1)
a020884_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             = u : f u (v + 2)
         where uu = u ^ 2; w = a037213 (uu + v ^ 2)
-- Reinhard Zumkeller, Nov 09 2012

Crossrefs

Cf. A009004, A020882, A020883, A020885, A020886. Different from A024352.

Cf. A024359 (gives the number of times n occurs).

Cf. A037213.

Sequence in context: A196115 A336867 A025051 * A183855 A024352 A288525

Adjacent sequences: A020881 A020882 A020883 * A020885 A020886 A020887

Keyword

nonn,easy,nice

Author

Clark Kimberling

Extensions

Extended and corrected by David W. Wilson

Status

approved