A046087 -id:A046087 - cp's OEIS frontend

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

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

A263728 Primitive Pythagorean triples (a, b, c) in lexicographic order, with a < b < c.

Original entry on oeis.org

3, 4, 5, 5, 12, 13, 7, 24, 25, 8, 15, 17, 9, 40, 41, 11, 60, 61, 12, 35, 37, 13, 84, 85, 15, 112, 113, 16, 63, 65, 17, 144, 145, 19, 180, 181, 20, 21, 29, 20, 99, 101, 21, 220, 221, 23, 264, 265, 24, 143, 145, 25, 312, 313, 27, 364, 365, 28, 45, 53

Offset: 1

Colin Barker, Nov 20 2015

a(3*k+1)*a(3*k+2) / (a(3*k+1)+a(3*k+2)+a(3*k+3)) is always an integer for k >= 0. Also note that a(3*k+1)*a(3*k+2)/2 is never a perfect square. - Altug Alkan, Apr 08 2016

			The first few triples are [3, 4, 5], [5, 12, 13], [7, 24, 25], [8, 15, 17], [9, 40, 41], [11, 60, 61], [12, 35, 37], [13, 84, 85], [15, 112, 113], [16, 63, 65], [17, 144, 145], [19, 180, 181], [20, 21, 29], [20, 99, 101], ... - _N. J. A. Sloane_, Dec 15 2015

H. M. Stark, An Introduction to Number Theory. Markham, Chicago, 1970, Chapter 5, Section 5.3.

Colin Barker, Table of n, a(n) for n = 1..10000
Yoshinosuke Hirakawa and Hideki Matsumura, A unique pair of triangles, Journal of Number Theory, Volume 194, January 2019, Pages 297-302. About the remarkable (135,352,377) Pythagorean triple.
Eric Weisstein's World of Mathematics, Pythagorean Triple
Wikipedia, Pythagorean triple

Cf. A008846, A020882, A046086, A046087, A103606, A139794, A156678, A156679.

A277557 The ordered image of the 1-to-1 mapping of an integer ordered pair (x,y) into an integer using Cantor's pairing function, where 0 < x < y, gcd(x,y)=1 and x+y odd.

Original entry on oeis.org

8, 18, 19, 32, 33, 34, 50, 52, 53, 72, 73, 74, 75, 76, 98, 99, 100, 101, 102, 103, 128, 131, 133, 134, 162, 163, 164, 165, 166, 167, 168, 169, 200, 201, 202, 203, 204, 205, 206, 207, 208, 242, 244, 247, 248, 250, 251, 288, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 338

Offset: 1

Frank M Jackson, Oct 19 2016

nonn

The mapping of the ordered pair (x,y) to an integer uses Cantor's pairing function to generate the integer as (x+y)(x+y+1)/2+y. Also for every ordered pair (x,y) such that 0 < x < y, gcd(x,y)=1 and x+y odd, there exists a primitive Pythagorean triple (PPT) (a, b, c) such that a = y^2-x^2, b = 2xy, c = x^2+y^2. Therefore each term in the sequence represents a unique PPT.
Numbers n for which 0 < A025581(n) < A002262(n) and A025581(n)+A002262(n) is odd, and gcd(A025581(n), A002262(n)) = 1. [The definition expressed with A-numbers.] - Antti Karttunen, Nov 02 2016
See also the triangle T(y, x) with the values for PPTs given in A278147. - Wolfdieter Lang, Nov 24 2016

			a(5)=33 because the ordered pair (2,5) maps to 33 by Cantor's pairing function (see below) and is the 5th such occurrence. Also x=2, y=5 generates a PPT with sides (21,20,29).
Note: Cantor's pairing function is simply A001477 in its two-argument tabular form A001477(k, n) = n + (k+n)*(k+n+1)/2, thus A001477(2,5) = 5 + (2+5)*(2+5+1)/2 = 33. - _Antti Karttunen_, Nov 02 2016

Wikipedia, Cantor's pairing function, and Pythagorean triple

Cf. A001477, A002262, A025581, A243808, A277632.
Cf. A020882 (is obtained when A048147(a(n)) is sorted into ascending order), A008846 (same with duplicates removed).
Cf. A020887, A020888, A120427, A024362, A024406, A046079, A046087, A070151, A156678, A156679, A156680, A156683, A156685, A222946, A278147.

Cantor[{i_, j_}] := (i+j)(i+j+1)/2+j; getparts[n_] := Reverse@Select[Reverse[IntegerPartitions[n, {2}], 2], GCD@@#==1 &]; pairs=Flatten[Table[getparts[2n+1], {n, 1, 20}], 1]; Table[Cantor[pairs[[n]]], {n, 1, Length[pairs]}]

A386308 Long legs of Pythagorean triples that do not have the form (u^2 - v^2, 2uv, u^2 + v^2) ordered by increasing hypotenuse (A386307), where u and v are positive integers.

Original entry on oeis.org

12, 20, 24, 28, 36, 40, 45, 44, 48, 52, 60, 56, 60, 72, 72, 68, 75, 63, 84, 76, 80, 90, 84, 88, 105, 92, 105, 96, 120, 120, 104, 120, 108, 112, 132, 105, 116, 120, 144, 124, 144, 135, 132, 156, 136, 150, 126, 140, 168, 168, 180, 148, 175, 165, 152, 156, 168, 180

Offset: 1

Felix Huber, Aug 19 2025

nonn

In the form (u^2 - v^2, 2*u*v, u^2 + v^2), u^2 + v^2 is the hypotenuse, max(u^2 - v^2, 2*u*v) is the long leg and min(u^2 - v^2, 2*u*v) is the short leg.

			The Pythagorean triple (9, 12, 15) does not have the form (u^2 - v^2, 2*u*v, u^2 + v^2), because 15 is not a sum of two nonzero squares. Therefore 12 is a term.

Felix Huber, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pythagorean Triple

Subsequence of A046084.

Cf. A366675, A380073, A386307, A386309, A386944.

A386308:=proc(N) # To get all terms with hypotenuses <= N
    local i,l,m,u,v,r,x,y,z;
    l:={};
    m:={};
    for u from 2 to floor(sqrt(N-1)) do
        for v to min(u-1,floor(sqrt(N-u^2))) do
            x:=min(2*u*v,u^2-v^2);
            y:=max(2*u*v,u^2-v^2);
            z:=u^2+v^2;
            m:=m union {[z,y,x]};
            if gcd(u,v)=1 and is(u-v,odd) then
                l:=l union {seq([i*z,i*y,i*x],i=1..N/z)}
            fi
        od
    od;
    r:=l minus m;
    return seq(r[i,2],i=1..nops(r));
end proc;
A386308(1000);

a(n) = sqrt(A386307(n)^2 - A386309(n)^2).

{A046084(n)} = {a(n)} union {A046087(n)} union {A386944(n)}.

A081872 Long legs of primitive Pythagorean triangles sorted on semiperimeter.

Original entry on oeis.org

4, 12, 15, 24, 21, 35, 40, 45, 60, 63, 56, 55, 84, 77, 80, 99, 72, 112, 91, 117, 144, 143, 105, 140, 132, 180, 165, 120, 176, 195, 153, 168, 220, 187, 156, 221, 171, 255, 208, 264, 209, 260, 247, 252, 285, 312, 231, 323, 240, 308, 273, 224, 364, 253, 357, 288

Offset: 1

Lekraj Beedassy, Apr 23 2003

nonn

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

Cf. A020886, A020883, A046087.

More terms from Ray Chandler, Oct 29 2003

A087459 Values (X + Y - Z) sorted on Z, then on Y, where (X,Y,Z) is a primitive Pythagorean triple with X

Original entry on oeis.org

2, 4, 6, 6, 12, 10, 8, 20, 10, 24, 14, 30, 28, 12, 30, 40, 18, 42, 14, 36, 56, 22, 16, 42, 60, 70, 44, 18, 72, 48, 70, 26, 84, 66, 90, 20, 52, 80, 88, 30, 78, 22, 60, 90, 110, 112, 60, 126, 104, 24, 66, 132, 34, 126, 130, 144, 68, 26, 154, 120, 110, 140, 156, 38, 102, 28

Offset: 1

Lekraj Beedassy, Oct 23 2003

nonn

Ray Chandler, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pythagorean Triple.

For ordered values of (X + Y - Z) see A020887.

Cf. A046086, A046087, A020882, A014498.

a(n) = A046086(n) + A046087(n) - A020882(n) = 2*A014498(n).

a(n) = sqrt{2*A118961(n)*A118962(n)}. - Lekraj Beedassy, May 11 2006

Corrected and extended by Ray Chandler, Oct 25 2003

A386944 Long legs of Pythagorean triples of the form (u^2 - v^2, 2uv, u^2 + v^2), ordered by increasing hypotenuse (A386943).

Original entry on oeis.org

8, 16, 24, 30, 32, 36, 48, 48, 42, 60, 70, 64, 80, 72, 96, 96, 90, 84, 108, 120, 100, 112, 126, 120, 110, 140, 135, 128, 160, 154, 168, 160, 144, 144, 192, 198, 192, 180, 182, 216, 224, 168, 216, 240, 196, 200, 234, 224, 252, 189, 240, 210, 286, 288, 220, 280, 280

Offset: 1

Felix Huber, Aug 24 2025

In the form (u^2 - v^2, 2*u*v, u^2 + v^2), u^2 + v^2 is the hypotenuse, max(u^2 - v^2, 2*u*v) is the long leg and min(u^2 - v^2, 2*u*v) is the short leg.

			The nonprimitive Pythagorean triple (6, 8, 10) is of the form (u^2 - v^2, 2*u*v, u^2 + v^2): From u = 3 and v = 1 follows u^2 - v^2 = 8 (long leg), 2*u*v = 6 (short leg), u^2 - v^2 = 10 (hypotenuse). Therefore, 8 is a term.

Felix Huber, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pythagorean Triple

Subsequence of A046084.

Cf. A046087, A366675, A380073, A386308, A386943, A386945.

A386944:=proc(N) # To get all hypotenuses <= N
    local i,l,u,v;
    l:=[];
    for u from 2 to floor(sqrt(N-1)) do
        for v to min(u-1,floor(sqrt(N-u^2))) do
            if gcd(u,v)>1 or is(u-v,even) then
                l:=[op(l),[u^2+v^2,max(2*u*v,u^2-v^2),min(2*u*v,u^2-v^2)]]
            fi
        od
    od;
    l:=sort(l);
    return seq(l[i,2],i=1..nops(l));
end proc;
A386944(296);

a(n) = sqrt(A386943(n)^2 - A386945(n)^2).

{A046084(n)} = {a(n)} union {A046087(n)} union {A386308(n)}.

A239581 Number of primitive Pythagorean triangles (x, y, z) with legs x < y < 10^n.

Original entry on oeis.org

1, 18, 179, 1788, 17861, 178600, 1786011, 17860355, 178603639, 1786036410, 17860362941

Offset: 1

Martin Renner, Mar 26 2014

A Pythagorean triangle is a right triangle with integer side lengths x, y, z forming a Pythagorean triple (x, y, z). It is called primitive, if gcd(x, y, z) = 1.

Because (x, y, z) is equivalent to (y, x, z), the total number of primitive Pythagorean triangles with legs x, y < 10^n is b(n) = 2*a(n) = 2, 36, 358, 3576, 35722, ...

			a(1) = 1, because the only primitive Pythagorean triangle with x < y < 10 is [3, 4, 5].

Eric Weisstein's World of Mathematics, Pythagorean Triangle.
Eric Weisstein's World of Mathematics, Pythagorean Triple.

Cf. A008846, A020882, A020883, A020884, A046086, A046087, A101931, A101929, A239744, A239786.

a(6)-a(11) from Giovanni Resta, Mar 27 2014

A239744 Number of Pythagorean triangles (x, y, z) with legs x < y <= 10^n.

Original entry on oeis.org

2, 63, 1034, 14474, 185864, 2269788, 26809924, 309224756, 3503496007, 39147452729, 432599522197

Offset: 1

Martin Renner, Mar 26 2014

A Pythagorean triangle is a right triangle with integer side lengths x, y, z forming a Pythagorean triple (x, y, z).

Because (x, y, z) is equivalent to (y, x, z), the total number of Pythagorean triangles with legs x, y < 10^n is b(n) = 2*a(n) = 4, 126, 2068, 28948, 371728, ...

			a(1) = 2, because the only two Pythagorean triangles with x < y < 10 are [3, 4, 5] and [6, 8, 10].

Eric Weisstein's World of Mathematics, Pythagorean Triangle.
Eric Weisstein's World of Mathematics, Pythagorean Triple.

Cf. A008846, A020882, A020883, A020884, A046086, A046087, A101931, A101929, A239581, A239786.

a(6)-a(11) from Giovanni Resta, Mar 27 2014

A239786 Number of Pythagorean triangles (x, y, z) with legs x < y < 10^n.

Original entry on oeis.org

2, 62, 1032, 14471, 185860, 2269783, 26809918, 309224749, 3503495999, 39147452720, 432599522187

Offset: 1

Martin Renner, Mar 26 2014

A Pythagorean triangle is a right triangle with integer side length x, y, z forming a Pythagorean triple (x, y, z).

Because (x, y, z) is equivalent to (y, x, z), the total number of Pythagorean triangles with legs x, y < 10^n is b(n) = 2*a(n) = 4, 124, 2064, 28942, ...

Eric Weisstein's World of Mathematics, Pythagorean Triangle.
Eric Weisstein's World of Mathematics, Pythagorean Triple.

Cf. A008846, A020882, A020883, A020884, A046086, A046087, A101931, A101929, A239581, A239744.

a(5)-a(11) from Giovanni Resta, Mar 27 2014

cp's OEIS Frontend

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

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A263728 Primitive Pythagorean triples (a, b, c) in lexicographic order, with a < b < c.

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A277557 The ordered image of the 1-to-1 mapping of an integer ordered pair (x,y) into an integer using Cantor's pairing function, where 0 < x < y, gcd(x,y)=1 and x+y odd.

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Mathematica

A386308 Long legs of Pythagorean triples that do not have the form (u^2 - v^2, 2*u*v, u^2 + v^2) ordered by increasing hypotenuse (A386307), where u and v are positive integers.

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Maple

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A081872 Long legs of primitive Pythagorean triangles sorted on semiperimeter.

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A087459 Values (X + Y - Z) sorted on Z, then on Y, where (X,Y,Z) is a primitive Pythagorean triple with X

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A386944 Long legs of Pythagorean triples of the form (u^2 - v^2, 2*u*v, u^2 + v^2), ordered by increasing hypotenuse (A386943).

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A239581 Number of primitive Pythagorean triangles (x, y, z) with legs x < y < 10^n.

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A239744 Number of Pythagorean triangles (x, y, z) with legs x < y <= 10^n.

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A386308 Long legs of Pythagorean triples that do not have the form (u^2 - v^2, 2uv, u^2 + v^2) ordered by increasing hypotenuse (A386307), where u and v are positive integers.

A386944 Long legs of Pythagorean triples of the form (u^2 - v^2, 2uv, u^2 + v^2), ordered by increasing hypotenuse (A386943).