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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.

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A297207 3d-congruent numbers: positive integers n for which there exists a trirectangular tetrahedron having volume n and rational areas and sides.

Original entry on oeis.org

13, 85, 104, 175, 308, 351, 490, 680, 832, 1400, 1625, 2295, 2464, 2808, 3179, 3920, 4459, 4725, 5440, 6656, 8316, 9295, 9477, 9724, 10625, 11200, 13000, 13230, 13475, 17303, 18360, 19712, 19760, 21875, 22464, 25432, 28561, 29155, 31360, 35672, 37800, 38500, 43520, 43875
Offset: 1

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Author

Ralf Steiner, Dec 27 2017

Keywords

Comments

Ordered list of numbers A295507(i)*j^3 with i,j positive integers.

Examples

			a(1) = 13 = 13*1^3 = A295507(1)*A000578(1) is the volume of the trirectangular tetrahedron with sides: {240, 117, 44, 125, 244, 267}*1^3/15840.
a(3) = 104 = 13*2^3 = A295507(1)*A000578(2) is the volume of the trirectangular tetrahedron with sides: {240, 117, 44, 125, 244, 267}*2^3/15840.

Links

Crossrefs

Cf. A295507, A000578 (cubes).

A297327 1/36 of the square of the basis of a primitive 3-simplex.

Original entry on oeis.org

6434041, 89002225, 865125625, 89610625, 353440516, 29160156025, 18989880481, 37434450625, 72399370000, 444515646025, 346008660625, 2003915162500, 9475360381201, 166729268761, 13110591519025, 8007417968121, 11201866562500, 3095696620900, 61956758281561
Offset: 1

Views

Author

Ralf Steiner, Dec 28 2017

Keywords

Comments

For every primitive trirectangular tetrahedron (0, a, b, c) with coprime integer sides, (b*c)^2 + (a*b)^2 + (c*a)^2 is divisible by 144.
The square of the basis is related by De Gua's theorem on the square of the main diagonal of a (different, not necessarily primitive) Euler brick (a*b/12=A031173(k), a*c/12=A031174(k), b*c/12=A031175(k)) also having integer sides and integer face diagonals including a trirectangular tetrahedron (0, a*b/12, a*c/12, b*c/12), such as a(1) = 6434041 = A023185(8) = A031173(8)^2 + A031174(8)^2 + A031175(8)^2.
By this process a cycle of primitive trirectangular tetrahedrons is defined, such as with indices k: (1 8), (2 6), (3 5), (4 7), (9 19), ...

Links

Crossrefs

Cf. A295507, A023185, A031173, A031174, A031175.

Formula

a(n) = (1/144)*(A031174(n)^2*A031175(n)^2 + A031173(n)^2*(A031174(n)^2 + A031175(n)^2)).
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