OFFSET
1,3
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, ...
LINKS
Eric Weisstein's World of Mathematics, Euler Brick
EXAMPLE
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.
PROG
(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
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Martin Renner, Mar 22 2014
EXTENSIONS
a(6)-a(8) from Giovanni Resta, Mar 22 2014
a(9) from Robin Visser, Jan 01 2024
STATUS
approved