A081874 -id:A081874 - cp's OEIS frontend

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

Clark Kimberling

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

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

Cf. A009004, A020882, A020883, A020885, A020886. Different from A024352.
Cf. A024359 (gives the number of times n occurs).
Cf. A037213.

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

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 *)
Take[Union[Sort/@({Times@@#,(Last[#]^2-First[#]^2)/2}&/@(Select[Subsets[Range[1,101,2],{2}],GCD@@#==1&]))][[;;,1]],80] (* Harvey P. Dale, Feb 06 2025 *)

Extended and corrected by David W. Wilson

A180620 Odd legs of primitive Pythagorean triples (with multiplicity) sorted with respect to increasing hypotenuse.

Original entry on oeis.org

3, 5, 15, 7, 21, 35, 9, 45, 11, 63, 33, 55, 77, 13, 39, 65, 99, 91, 15, 117, 105, 143, 17, 51, 85, 119, 165, 19, 153, 57, 95, 195, 187, 133, 171, 21, 221, 105, 209, 255, 247, 23, 69, 115, 231, 161, 285, 273, 207, 25, 75, 323, 253, 175, 299, 225, 357, 27, 275, 345, 135, 189, 325

Offset: 1

Jonathan Vos Post, Sep 12 2010

The primary key is the increasing length of the hypotenuse, A020882. If there is more than one solution with that hypotenuse, the (secondary) sorting key is the even leg.

Only the odd legs 'a' of reduced triangles with gcd(a,b,c)=1, a^2+b^2=c^2, a=q^2-p^2, b=2*p*q, c=q^2+p^2, gcd(p,q)=1 are listed.

			a(1) = 3 because the only triangle with the least possible hypotenuse 5 has catheti 3 and 4.

K. G. Stier, Table of n, a(n) for n = 1..1593
Michael Somos, Pythagorean Triple Table, Reduced integer right triangles, Feb 28, 1998.
Wikipedia, Pythagorean Triple.
Index entries related to Pythagorean Triples.

Cf. A046086, A046087, A020882, A081874, A231100.

Comment on sorting added, more terms appended by R. J. Mathar, Oct 15 2010

Sequence's name and comments corrected by K. G. Stier, Nov 03 2013

A120890 Ordered odd leg of primitive Pythagorean triangles.

Original entry on oeis.org

3, 5, 7, 9, 11, 13, 15, 15, 17, 19, 21, 21, 23, 25, 27, 29, 31, 33, 33, 35, 35, 37, 39, 39, 41, 43, 45, 45, 47, 49, 51, 51, 53, 55, 55, 57, 57, 59, 61, 63, 63, 65, 65, 67, 69, 69, 71, 73, 75, 75, 77, 77, 79, 81, 83, 85, 85, 87, 87, 89, 91, 91, 93, 93, 95, 95, 97, 99, 99, 101, 103

Offset: 1

Lekraj Beedassy, Jul 12 2006

nonn

Ordered union of A081874 and A081934.

Conjecture: lim_{n->oo} a(n)/n = 1/Pi. Limit is also conjectured to be equal to lim_{n->oo} A120427(n)/n, see Selle reference, chapter 2.3.10. - Lothar Selle, Jun 21 2022

Lothar Selle, Kleines Handbuch Pythagoreische Zahlentripel, Books on Demand, 4th impression 2022, chapter 2.2.1., see chapter 2.3.10 for identity of lim_{n->oo} A120427(n)/n.

Ray Chandler, Table of n, a(n) for n = 1..10000
D. N. Lehmer, Asymptotic evaluation of certain totient sums, Amer. J. Math. 22, 293-335, 1900.
F. Richman, Pythagorean Triangles

Cf. A061346, A081874, A081934, A120891.

Cf. A120427.

Corrected by T. D. Noe, Oct 25 2006

A112398 Odd numbers which are exclusively the shortest side of primitive Pythagorean triangles.

Original entry on oeis.org

3, 7, 9, 11, 19, 23, 27, 31, 33, 39, 43, 47, 49, 51, 57, 59, 67, 69, 71, 75, 79, 81, 83, 87, 93, 95, 103, 107, 111, 115, 119, 121, 123, 127, 129, 131, 133, 135, 139, 141, 147, 151, 155, 159, 161, 163, 167, 175, 177, 179, 183, 189, 191, 199, 201, 203, 207, 211, 213

Offset: 1

Lekraj Beedassy, Dec 06 2005

nonn

Numbers that are in A081874 (odd short legs) but in neither A081934 (odd long legs) nor A020882 (hypotenuse).

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

Entries of A112372 excluding entries in A020882.

Extended by Ray Chandler, Dec 08 2005

A083883 Odd short leg of more than one primitive Pythagorean triangle.

Original entry on oeis.org

33, 39, 51, 57, 65, 69, 75, 85, 87, 93, 95, 105, 111, 115, 119, 123, 129, 133, 135, 141, 145, 147, 155, 159, 161, 165, 175, 177, 183, 185, 189, 195, 201, 203, 205, 207, 213, 215, 217, 219, 225, 231, 235, 237, 245, 249, 255, 259, 261, 265, 267, 273, 279, 285

Offset: 1

Lekraj Beedassy, Jun 18 2003

nonn

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

Cf. A081874.

Take[Transpose[Select[Tally[Transpose[Union[Sort/@({Times@@#,(Last[#]^2- First[#]^2)/2}&/@(Select[Subsets[Range[1,501,2],{2}],GCD@@#==1&]))]][[1]]], OddQ[#[[1]]]&&#[[2]]>1&]][[1]],60] (* Harvey P. Dale, May 28 2015 *)

Corrected and extended by Ray Chandler, Oct 29 2003

A112372 Odd legs of primitive Pythagorean triangles which are exclusively the short one.

Original entry on oeis.org

3, 5, 7, 9, 11, 13, 17, 19, 23, 25, 27, 29, 31, 33, 37, 39, 41, 43, 47, 49, 51, 53, 57, 59, 61, 65, 67, 69, 71, 73, 75, 79, 81, 83, 85, 87, 89, 93, 95, 97, 101, 103, 107, 109, 111, 113, 115, 119, 121, 123, 125, 127, 129, 131, 133, 135, 137, 139, 141, 145, 147, 149, 151

Offset: 1

Lekraj Beedassy, Dec 02 2005

nonn

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

Entries of A081874 (removing duplicates) excluding its subsequence A081934.

Cf. A112398.

Extended by Ray Chandler, Dec 08 2005

cp's OEIS Frontend

A020884 Ordered short legs of primitive Pythagorean triangles.

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A180620 Odd legs of primitive Pythagorean triples (with multiplicity) sorted with respect to increasing hypotenuse.

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A120890 Ordered odd leg of primitive Pythagorean triangles.

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A112398 Odd numbers which are exclusively the shortest side of primitive Pythagorean triangles.

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A083883 Odd short leg of more than one primitive Pythagorean triangle.

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A112372 Odd legs of primitive Pythagorean triangles which are exclusively the short one.

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