A386307 Ordered hypotenuses of Pythagorean triples that do not have the form (u^2 - v^2, 2*u*v, u^2 + v^2), where u and v are positive integers.

15, 25, 30, 35, 39, 50, 51, 55, 60, 65, 65, 70, 75, 75, 78, 85, 85, 87, 91, 95, 100, 102, 105, 110, 111, 115, 119, 120, 123, 125, 130, 130, 135, 140, 143, 145, 145, 150, 150, 155, 156, 159, 165, 169, 170, 170, 174, 175, 175, 182, 183, 185, 185, 187, 190, 195, 195

Offset: 1

Author

Felix Huber, Aug 13 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.

A101930(n) gives the total number of Pythagorean triples <= 10^n. The percentage of triangles in this sequence increases continuously:

number of terms <= h total number of

h in this sequence hypotenuses <= h percentage

10 0 2 0.0 %

100 21 52 40.4 %

1000 514 881 58.3 %

10000 8629 12471 69.2 %

100000 122431 161436 75.8 %

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 15 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 A009000.

Cf. A101930, A366428, A380072, A386308, A386309, A386943.

Programs

Maple

A386307:=proc(N) # To get all 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,1],i=1..nops(r));
end proc;
A386307(1000);

Formula

a(n) = sqrt(A386308(n)^2 + A386309(n)^2).

{A009000(n)} = {a(n)} union {A020882(n)} union {A386943(n)}.