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A239618
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Number of primitive Euler bricks with side length a < b < c < 10^n, i.e., in a boxed parameter space with dimension 10^n.
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1
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OFFSET
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1,3
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COMMENTS
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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, ...
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LINKS
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EXAMPLE
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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.
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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