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.

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

Author

Felix Huber, Aug 19 2025

Keywords

• nonn
• new

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

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

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)}.