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A031173
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Longest edge a of smallest (measured by the longest edge) primitive Euler bricks (a, b, c, sqrt(a^2 + b^2), sqrt(b^2 + c^2), sqrt(a^2 + c^2) are integers).
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20
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240, 275, 693, 720, 792, 1155, 1584, 2340, 2640, 2992, 3120, 5984, 6325, 6336, 6688, 6732, 8160, 9120, 9405, 10725, 11220, 12075, 13860, 14560, 16800, 17472, 17748, 18560, 19305, 21476, 23760, 23760, 24684, 25704, 26649, 29920, 30780
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OFFSET
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1,1
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COMMENTS
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Primitive means that gcd(a,b,c) = 1.
The trirectangular tetrahedron (0, a=a(n), b=A031174(n), c=A031175(n)) has three right triangles with area divisible by 6 = 2*3 each and a volume divisible by 15840 = 2^5*3^2*5*11. The biquadratic term b^2*c^2 + a^2*(b^2 + c^2) is divisible by 144 = 2^4*3^2. Also gcd(b + c, c + a, a + b) = 1. - Ralf Steiner, Nov 22 2017
There are some longest edges a which occur multiple times, such as a(31) = a(32) = 23760. - Ralf Steiner, Jan 07 2018
A trirectangular tetrahedron is never a perfect body (in the sense of Wyss) because it always has an irrational area of the base (a,b,c) whose value is half of the length of the space-diagonal of the related cuboid (b*c, c*a, a*b). The trirectangular bipyramid (6 faces, 9 edges, 5 vertices) built from these trirectangular tetrahedrons and the related left-handed ones connected on their bases have rational numbers for volume, face areas and edge lengths, but again an irrational value for the length of the space-diagonal which is a rational part of the length of the space-diagonal of the related cuboid (b*c, c*a, a*b). - Ralf Steiner, Jan 14 2018
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REFERENCES
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Calculated by F. Helenius (fredh(AT)ix.netcom.com).
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LINKS
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CROSSREFS
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KEYWORD
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nonn,changed
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AUTHOR
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STATUS
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approved
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