This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A239618 #21 Feb 16 2025 08:33:21 %S A239618 0,0,5,19,65,242,704,1884,4631 %N 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. %C A239618 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. %C A239618 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, ... %H A239618 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EulerBrick.html">Euler Brick</a> %H A239618 <a href="/index/Br#bricks">Index entries for sequences related to bricks</a> %e A239618 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. %o A239618 (Sage) %o A239618 def a(n): %o A239618 ans = 0 %o A239618 for x in range(1,10^n): %o A239618 divs = Integer(x^2).divisors() %o A239618 for d in divs: %o A239618 if (d <= x^2/d): continue %o A239618 if (d-x^2/d >= 2*x): break %o A239618 if (d-x^2/d)%2==0: %o A239618 y = (d-x^2/d)/2 %o A239618 for e in divs: %o A239618 if (e <= x^2/e): continue %o A239618 if (e-x^2/e >= 2*y): break %o A239618 if (e-x^2/e)%2==0: %o A239618 z = (e-x^2/e)/2 %o A239618 if (gcd([x,y,z])==1) and (y^2+z^2).is_square(): %o A239618 ans += 1 %o A239618 return ans # _Robin Visser_, Jan 01 2024 %Y A239618 Cf. A031173, A031174, A031175, A239620. %K A239618 nonn,more %O A239618 1,3 %A A239618 _Martin Renner_, Mar 22 2014 %E A239618 a(6)-a(8) from _Giovanni Resta_, Mar 22 2014 %E A239618 a(9) from _Robin Visser_, Jan 01 2024