A239618 Number of primitive Euler bricks with side length a < b < c < 10^n, i.e., in a boxed parameter space with dimension 10^n.

0, 0, 5, 19, 65, 242, 704, 1884, 4631

Offset: 1

Author

Martin Renner, Mar 22 2014

Keywords

• nonn
• more

Comments

An Euler brick is a cuboid of integer side dimensions a, b, c such that the face diagonals are integers. It is called primitive if gcd(a,b,c)=1.

Because the sides of a cuboid are permutable without changing its shape, the total number of primitive Euler bricks in the parameter space a, b, c < 10^n is b(n) = 6*a(n) = 0, 0, 30, 114, 390, ...

Examples

			a(3) = 5, since there are the five primitive Euler bricks [44, 117, 240], [85, 132, 720], [140, 480, 693], [160, 231, 792], [240, 252, 275] with longest side length < 1000.
		

Links

• Eric Weisstein's World of Mathematics, Euler Brick
• Index entries for sequences related to bricks

Crossrefs

Cf. A031173, A031174, A031175, A239620.

Programs

Sage

def a(n):
    ans = 0
    for x in range(1,10^n):
        divs = Integer(x^2).divisors()
        for d in divs:
            if (d <= x^2/d): continue
            if (d-x^2/d >= 2*x): break
            if (d-x^2/d)%2==0:
                y = (d-x^2/d)/2
                for e in divs:
                    if (e <= x^2/e): continue
                    if (e-x^2/e >= 2*y): break
                    if (e-x^2/e)%2==0:
                        z = (e-x^2/e)/2
                        if (gcd([x,y,z])==1) and (y^2+z^2).is_square():
                            ans += 1
    return ans  # Robin Visser, Jan 01 2024

Extensions

a(6)-a(8) from Giovanni Resta, Mar 22 2014

a(9) from Robin Visser, Jan 01 2024