cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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.

Original entry on oeis.org

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

Views

Author

Felix Huber, Aug 19 2025

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

Subsequence of A046083.
Cf. A366674, A380074, A386307, A386308, A386945.

Programs

  • Maple
    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);

Formula

a(n) = sqrt(A386307(n)^2 - A386308(n)^2).
{A046083(n)} = {a(n)} union {A046086(n)} union {A386945(n)}.

A386307 Ordered hypotenuses of Pythagorean triples that do not have the form (u^2 - v^2, 2*u*v, u^2 + v^2), where u and v are positive integers.

Original entry on oeis.org

15, 25, 30, 35, 39, 50, 51, 55, 60, 65, 65, 70, 75, 75, 78, 85, 85, 87, 91, 95, 100, 102, 105, 110, 111, 115, 119, 120, 123, 125, 130, 130, 135, 140, 143, 145, 145, 150, 150, 155, 156, 159, 165, 169, 170, 170, 174, 175, 175, 182, 183, 185, 185, 187, 190, 195, 195
Offset: 1

Views

Author

Felix Huber, Aug 13 2025

Keywords

Comments

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.
A101930(n) gives the total number of Pythagorean triples <= 10^n. The percentage of triangles in this sequence increases continuously:
number of terms <= h total number of
h in this sequence hypotenuses <= h percentage
10 0 2 0.0 %
100 21 52 40.4 %
1000 514 881 58.3 %
10000 8629 12471 69.2 %
100000 122431 161436 75.8 %

Examples

			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 15 is a term.

Links

Crossrefs

Subsequence of A009000.
Cf. A101930, A366428, A380072, A386308, A386309, A386943.

Programs

  • Maple
    A386307:=proc(N) # To get all 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,1],i=1..nops(r));
end proc;
A386307(1000);

Formula

a(n) = sqrt(A386308(n)^2 + A386309(n)^2).
{A009000(n)} = {a(n)} union {A020882(n)} union {A386943(n)}.

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).

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

Views

Author

Felix Huber, Aug 24 2025

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

Subsequence of A046084.
Cf. A046087, A366675, A380073, A386308, A386943, A386945.

Programs

  • Maple
    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);

Formula

a(n) = sqrt(A386943(n)^2 - A386945(n)^2).
{A046084(n)} = {a(n)} union {A046087(n)} union {A386308(n)}.
Showing 1-3 of 3 results.

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