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

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

1,1

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.

Links

Felix Huber, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Pythagorean Triple

Formula

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

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

Example

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.

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);

Crossrefs

Subsequence of A046084.

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

Sequence in context: A177899 A358727 A246311 * A344538 A244371 A389212

Adjacent sequences: A386941 A386942 A386943 * A386945 A386946 A386947

Keyword

nonn,easy

Author

Felix Huber, Aug 24 2025

Status

approved