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.

A366428 Hypotenuse numbers w of Pythagorean triples (u, v, w) for which (u^2, v^2, w^2) is an "abc-hit".

Original entry on oeis.org

25, 41, 65, 125, 145, 289, 337, 377, 425, 625, 677, 841, 845, 1025, 1201, 1625, 1681, 1985, 2125, 2197, 2305, 2873, 3125, 3281, 3425, 3721, 4097, 4225, 4481, 4705, 4825, 4901, 4913, 5329, 6401, 6625, 6725, 6845, 7585, 7813, 7817, 8065, 8177, 9409, 10625, 10985
Offset: 1

Views

Author

Felix Huber, Oct 13 2023

Keywords

Comments

(a, b, c) is an ABC triple if gcd(a, b) = 1 and a + b = c. ABC triples with c > rad(a*b*c) are called "abc-hits". For primitive Pythagorean triples (u, v, w) it is u^2 + v^2 = w^2 and gcd(u^2, v^2) = 1. (u^2, v^2, w^2) are therefore ABC triples. They are then "abc-hits" if in addition w^2 > rad(u^2*v^2*w^2). If (u, v, w) is a non-primitive Pythagorean triple, (u^2, v^2, w^2) is not an ABC triple.
The corresponding values of min(u, v) and max(u, v) are in the sequences A366674 and A366675.
w of primitive Pythagorean triples (u, v, w) with A007947(u^2*v^2*w^2) < w^2.
Subsequence of intersection of A020882 and sqrt(A130510).

Examples

			25 from the primitive Pythagorean triple (7, 24, 25) is in the sequence, because 7^2 + 24^2 = 25^2, gcd(7^2, 24^2) = 1 and 25^2 = 625 > rad(7^2*24^2*25^2) = 7*2*3*5 = 210.

Links

Crossrefs

Cf. A366674, A366675 (corresponding values of min(u, v) and max(u, v)).
Cf. A020882 (hypotenuses of primitive Pythagorean triangles), A130510 ("abc-hits"), A007947 (squarefree kernel).

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.

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

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 A046084.
Cf. A366675, A380073, A386307, A386309, A386944.

Programs

  • Maple
    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);

Formula

a(n) = sqrt(A386307(n)^2 - A386309(n)^2).
{A046084(n)} = {a(n)} union {A046087(n)} union {A386944(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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