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

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

1,1

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 - A386944(n)^2).

{A046083(n)} = {a(n)} union {A046086(n)} union {A386309(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, 6 is a term.

Maple

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);

Crossrefs

Subsequence of A046083.

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

Sequence in context: A352482 A164378 A124349 * A360719 A365787 A315566

Adjacent sequences: A386942 A386943 A386944 * A386946 A386947 A386948

Keyword

nonn,easy

Author

Felix Huber, Aug 24 2025

Status

approved