A046086 -id:A046086 - cp's OEIS frontend

A386309 Short 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.

9, 15, 18, 21, 15, 30, 24, 33, 36, 39, 25, 42, 45, 21, 30, 51, 40, 60, 35, 57, 60, 48, 63, 66, 36, 69, 56, 72, 27, 35, 78, 50, 81, 84, 55, 100, 87, 90, 42, 93, 60, 84, 99, 65, 102, 80, 120, 105, 49, 70, 33, 111, 60, 88, 114, 117, 99, 75, 48, 120, 140, 96, 123, 45

Offset: 1

Felix Huber, Aug 19 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 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 A046083.

Cf. A366674, A380074, A386307, A386308, A386945.

A386309:=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,3],i=1..nops(r));
end proc;
A386309(1000);

a(n) = sqrt(A386307(n)^2 - A386308(n)^2).

{A046083(n)} = {a(n)} union {A046086(n)} union {A386945(n)}.

A081859 Short leg of primitive Pythagorean triangles sorted on semiperimeter.

Original entry on oeis.org

3, 5, 8, 7, 20, 12, 9, 28, 11, 16, 33, 48, 13, 36, 39, 20, 65, 15, 60, 44, 17, 24, 88, 51, 85, 19, 52, 119, 57, 28, 104, 95, 21, 84, 133, 60, 140, 32, 105, 23, 120, 69, 96, 115, 68, 25, 160, 36, 161, 75, 136, 207, 27, 204, 76, 175, 180, 40, 225, 135, 29, 152, 252, 189, 120

Offset: 1

Lekraj Beedassy, Apr 23 2003

nonn

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

Cf. A020886, A020884, A046086.

More terms from Ray Chandler, Oct 28 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

A386945 Short legs of Pythagorean triples of the form (p^2 - q^2, 2pq, p^2 + q^2), ordered by increasing hypotenuse (A386943).

Original entry on oeis.org

6, 12, 10, 16, 24, 27, 14, 20, 40, 32, 24, 48, 18, 54, 28, 40, 56, 80, 45, 22, 75, 66, 32, 64, 96, 48, 72, 96, 36, 72, 26, 78, 108, 130, 56, 40, 80, 112, 120, 63, 30, 160, 90, 44, 147, 150, 88, 132, 64, 180, 128, 176, 48, 34, 192, 96, 102, 144, 170, 192, 125, 72

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, 6 is a term.

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

Subsequence of A046083.

Cf. A046086, A366674, A380074, A386309, A386943, A386944.

A386945:=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,3],i=1..nops(l));
end proc;
A386945(296);

a(n) = sqrt(A386943(n)^2 - A386944(n)^2).

{A046083(n)} = {a(n)} union {A046086(n)} union {A386309(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

A349536 Consider a circle on the Z X Z lattice with radius equal to the Pythagorean hypotenuse h(n) (A009003); a(n) = number of Pythagorean triples inside a Pi/4 sector of the circle.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 31, 32, 33, 34, 35, 37, 38, 39, 40, 44, 45, 46, 47, 48, 49, 50, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 71, 75, 76, 77, 78, 79, 80, 84, 85, 86, 87, 89

Offset: 1

Alexander Kritov, Nov 21 2021

nonn

Number of Pythagorean triples with hypotenuse less than or equal to the next one.

			The count of non-primitive Pythagorean triples as they appear in order of increasing hypotenuse:
.
       Hypotenuse
   n  (A009003(n))       Sides       a(n)
  --  ------------  ---------------  ----
   1        5            (3,4)         1
   2       10            (6,8)         2
   3       13            (5,12)        3
   4       15            (9,12)        4
   5       17            (8,15)        5
   6       20           (12,16)        6
   7       25       (7,24), (15,20)    8
   8       26           (10,24)        9
   9       29           (20,21)       10

W. Sierpinski, Pythagorean Triangles, Dover Publications, 2003.

Alexander Kritov, Table of n, a(n) for n = 1..1050
Manuel Benito and Juan L. Varona, Pythagorean triangles with legs less than n, Journal of Computational and Applied Mathematics 143, (2002), pp. 117-126.
E. Eckert, The group of primitive Pythagorean triangles, Mathematics Magazine 57 (1984) 22-27.
Alexander Kritov, C code that generates b-file
Eric Weisstein's World of Mathematics, Pythagorean Triple

Cf. A008846, A009003.
Cf. A020882, A081804, A008846, A020883, A046086.
Cf. A349538 (extension to the full circle of Z^2 lattice).

// see enclosed main.c
for (long j=1;j< 101;++j)
{
for (long k=1;k< 101;++k)
{
if (k<=j)   // to avoid pairs (as we need 1/8 or quarter plane)
    {
          double hyp=sqrt(j*j+k*k);
          double c= (double) floor (hyp );
if   (fabs(hyp - c) < DBL_EPSILON)  arr[r++]= (long) c;
}}}
bubbleSort(arr, r);//sort by hypotenuse increase
for (long j=0;j< r;++j)
{
   if  ( arr[j] != arr[j+1] )
    {
        // write to file: j is the sequence value a[n]*2
        // arr[j] is the hypotenuse value
    }
}

Conjecture: the increment is a(n+1) - a(n) = 2^(m-1), where m is the sum of all powers of the Pythagorean primes (A002144) in the factorization of hypotenuse h(n+1) (see Eckert for PPT). However, starting from 58 the increment is 3.

A120644 Area common to integer-sided isosceles triangles (x,x,y) and (x,x,z=y+2d), sorted on x > z/2, d being positive.

Original entry on oeis.org

12, 60, 120, 168, 420, 420, 360, 1260, 660, 1848, 1008, 2640, 2772, 1092, 3120, 4680, 1980, 5460, 1680, 5148, 9240, 3432, 2448, 7140, 11220, 14280, 8580, 3420, 15912, 10032, 15960, 5460, 20748, 15708, 23940, 4620, 13260, 21840, 25080, 8160, 23712

Offset: 1

Lekraj Beedassy, Aug 17 2006, Aug 20 2006

x=A020882(n); y=2*A046086(n); z=2*A046087(n); d=A120682(n). y is twice the height of the other triangle with z as base and conversely.

Take the n-th primitive Pythagorean triple (x, y, z) ordered by increasing z, then y. (1/x)^2 + (1/y)^2 = (z/w)^2, where a(n) = w. - Ivan N. Ianakiev, Jan 12 2020

			168 in the sequence refers to the area common to both triangle (25,25,14) and triangle (25,25,48).

a(n) = y*z/4 = A046086(n)*A046087(n) = 2*A120734(n).

A297878 1/4 of the even edges of primitive Pythagorean triangles with legs (b=A081872, c=A081859), ordered by semiperimeters.

Original entry on oeis.org

1, 3, 2, 6, 5, 3, 10, 7, 15, 4, 14, 12, 21, 9, 20, 5, 18, 28, 15, 11, 36, 6, 22, 35, 33, 45, 13, 30, 44, 7, 26, 42, 55, 21, 39, 15, 35, 8, 52, 66, 30, 65, 24, 63, 17, 78, 40, 9, 60, 77, 34, 56, 91, 51, 19, 72, 45, 10, 68, 88, 105, 38, 63, 85, 30, 104, 57, 21, 102, 120, 11, 76, 99, 42, 119, 70, 33, 95

Offset: 1

Ralf Steiner, Jan 07 2018

nonn

It seems that all positive integers are included.

Every term has the form of edge length e = (v-u)*u/2, semiperimeter s = (h+b+c)/2 = u*v with b > c, h^2 = b^2 + c^2, u < v < 2*u, v odd (see Theorem 3 of Witcosky).

			From _Michel Marcus_, Mar 07 2018: (Start)
The first 10 terms of A081859 are 3,  5,  8,  7, 20, 12,  9, 28, 11, 16;
The first 10 terms of A081872 are 4, 12, 15, 24, 21, 35, 40, 45, 60, 63;
So the first 10 even legs are     4, 12,  8, 24, 20, 12, 40, 28, 60, 16;
So the first 10 terms are         1,  3,  2,  6,  5,  3, 10,  7, 15,  4. (End)

Wikipedia, Pythagorean triple.
Lindsey Witcosky, Perimeters of primitive Pythagorean triangles.
Index entries related to Pythagorean Triples.

Cf. A298042((odd edge - 1)/2), A081872(b), A081859(c).

Cf. A231100 (even legs ordered by hypotenuse).

(* lists a0* have to be prepared before *)
opPT = {a020882, a046087, a046086, a020882 + a046087 + a046086} topPT = Transpose[opPT]; stopPT = SortBy[topPT, {#[[4]]} &]; tstopPT = Transpose[stopPT]; nopPT = tstopPT; Do[ If[OddQ[tstopPT[[2]][[k]]], nopPT[[2]][[k]] = tstopPT[[2]][[k]]; nopPT[[3]][[k]] = tstopPT[[3]][[k]], nopPT[[2]][[k]] = tstopPT[[3]][[k]]; nopPT[[3]][[k]] = tstopPT[[2]][[k]]], {k, 1, 10000}]; nopPT[[3]]/4

cp's OEIS Frontend

A386309 Short 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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

A081859 Short leg of primitive Pythagorean triangles sorted on semiperimeter.

Views

Author

Keywords

Links

Crossrefs

Extensions

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

Views

Author

Keywords

Links

Crossrefs

Formula

Extensions

A386945 Short legs of Pythagorean triples of the form (p^2 - q^2, 2*p*q, p^2 + q^2), ordered by increasing hypotenuse (A386943).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A349536 Consider a circle on the Z X Z lattice with radius equal to the Pythagorean hypotenuse h(n) (A009003); a(n) = number of Pythagorean triples inside a Pi/4 sector of the circle.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

C

Formula

A120644 Area common to integer-sided isosceles triangles (x,x,y) and (x,x,z=y+2d), sorted on x > z/2, d being positive.

Views

Author

Keywords

A386309 Short 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.

A386945 Short legs of Pythagorean triples of the form (p^2 - q^2, 2pq, p^2 + q^2), ordered by increasing hypotenuse (A386943).