A386945 Short legs of 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

Author

Felix Huber, Aug 24 2025

Keywords

• nonn
• easy
• 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 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.
		

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

Programs

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);

Formula

a(n) = sqrt(A386943(n)^2 - A386944(n)^2).

{A046083(n)} = {a(n)} union {A046086(n)} union {A386309(n)}.