A031173 -id:A031173 - cp's OEIS frontend

A298044 1/16 of the edge with the largest 2-adic valuation of a primitive 3-simplex (0, b=A031173, c=A031174, d=A031175).

15, 15, 30, 45, 10, 63, 99, 55, 52, 187, 195, 374, 33, 396, 418, 286, 510, 570, 572, 490, 221, 63, 273, 910, 1050, 1092, 660, 1160, 882, 885, 1148, 1485, 1505, 1092, 495, 1870, 308, 858, 1485, 585, 990, 385, 1333, 1001, 2907, 3045, 2277, 1394, 1475, 2310, 3690, 594, 3885, 3465, 2755, 1496, 4422, 2730, 3393

Offset: 1

Ralf Steiner, Jan 11 2018

nonn

Each primitive 3-simplex (0,b,c,d) with b > c > d is built by an odd, an even and an edge with the largest 2-adic valuation.

Eric Weisstein's World of Mathematics, Trirectangular Tetrahedron

Cf. A298046, A298047, A031173, A031174, A031175.

A298046 1/4 of the even edge of least 2-adic valuation of a primitive 3-simplex (0, b=A031173, c=A031174, d=A031175).

Original entry on oeis.org

11, 63, 35, 33, 198, 275, 255, 585, 660, 195, 207, 390, 1449, 187, 1575, 1683, 1222, 418, 741, 2457, 2805, 275, 3465, 270, 2574, 2205, 4437, 1518, 3850, 5369, 5940, 1225, 6171, 6426, 3808, 1950, 7695, 1890, 8901, 8976, 9275, 741, 6435, 297, 10395, 4615, 12831, 12870, 13299, 2133, 13570, 7843, 10593, 4901

Offset: 1

Ralf Steiner, Jan 11 2018

nonn

Each primitive 3-simplex (0,b,c,d) with b > c > d is built by 1 odd edge and 2 even edges; the edge that is considered here is the even one with the least 2-adic valuation.

Eric Weisstein's World of Mathematics, Trirectangular Tetrahedron

Cf. A298044, A298047, A031173, A031174, A031175.

A298047 The odd edge of a primitive 3-simplex (0, b=A031173, c=A031174, d=A031175).

Original entry on oeis.org

117, 275, 693, 85, 231, 1155, 187, 429, 855, 2475, 2035, 2295, 6325, 195, 1155, 1755, 495, 1575, 9405, 10725, 2925, 12075, 4901, 1881, 11753, 7579, 8789, 16929, 19305, 5643, 4599, 17157, 6435, 935, 26649, 23751, 10725, 35321, 35075, 1105, 2163, 38475, 38571, 39195, 5491, 15939, 51205, 24225, 9405, 57275

Offset: 1

Ralf Steiner, Jan 11 2018

nonn

Each primitive 3-simplex (0,b,c,d) with b > c > d is built by 1 odd edge and 2 even edges; the edge that is considered here is the odd edge.

Eric Weisstein's World of Mathematics, Trirectangular Tetrahedron

Cf. A298044, A298046, A031173, A031174, A031175.

A031174 Intermediate edge b 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).

Original entry on oeis.org

117, 252, 480, 132, 231, 1100, 1020, 880, 855, 2475, 2035, 2295, 5796, 748, 6300, 4576, 4888, 1672, 9152, 9828, 3536, 1100, 4901, 1881, 11753, 8820, 10560, 16929, 15400, 14160, 18368, 17157, 24080, 17472, 15232, 23751, 10725, 13728

Offset: 1

Eric W. Weisstein

nonn

Primitive means that gcd(a,b,c) = 1.

See A031173 for a list of the 3356 primitive bricks with c < b < a < 5*10^8. - Giovanni Resta, Mar 23 2014

Calculated by F. Helenius (fredh(AT)ix.netcom.com).

Giovanni Resta, Table of n, a(n) for n = 1..3556
R. R. Gallyamov, I. R. Kadyrov, D. D. Kashelevskiy, A fast modulo primes algorithm for searching perfect cuboids and its implementation, arXiv preprint arXiv:1601.00636 [math.NT], 2016.
A. A. Masharov and R. A. Sharipov, A strategy of numeric search for perfect cuboids in the case of the second cuboid conjecture, arXiv preprint, 2015.
J. Ramsden and H. Sharipov, Inverse problems associated with perfect cuboids, arXiv preprint arXiv:1207.6764, 2012. - From _N. J. A. Sloane_, Dec 23 2012
J. Ramsden and H. Sharipov, On singularities of the inverse problems associated with perfect cuboids, arXiv preprint arXiv:1208.1859, 2012. - From _N. J. A. Sloane_, Dec 25 2012
Ruslan Sharipov, Perfect cuboids and irreducible polynomials, arXiv:1108.5348, 2011
Ruslan Sharipov, A note on the first cuboid conjecture, arXiv:1109.2534, 2011
Ruslan Sharipov, A note on the second cuboid conjecture. Part I, arXiv:1201.1229, 2012
Ruslan Sharipov, Perfect cuboids and multisymmetric polynomials, arXiv preprint arXiv:1205.3135, 2012. - From _N. J. A. Sloane_, Oct 22 2012
Ruslan Sharipov, On an ideal of multisymmetric polynomials associated with perfect cuboids, arXiv preprint arXiv:1206.6769, 2012. - From _N. J. A. Sloane_, Dec 17 2012
Ruslan Sharipov, On the equivalence of cuboid equations and their factor equations, arXiv preprint arXiv:1207.2102, 2012. - From _N. J. A. Sloane_, Dec 22 2012
Ruslan Sharipov, A biquadratic Diophantine equation associated with perfect cuboids, arXiv preprint arXiv:1207.4081, 2012. - From _N. J. A. Sloane_, Dec 23 2012
Ruslan Sharipov, On a pair of cubic equations associated with perfect cuboids, arXiv preprint arXiv:1208.0308, 2012. - From _N. J. A. Sloane_, Dec 23 2012
Ruslan Sharipov, On two elliptic curves associated with perfect cuboids, arXiv preprint arXiv:1208.1227, 2012. - From _N. J. A. Sloane_, Dec 24 2012
Ruslan Sharipov, Asymptotic estimates for roots of the cuboid characteristic equation in the linear region, arXiv preprint, 2015.
Ruslan Sharipov, Reverse asymptotic estimates for roots of the cuboid characteristic equation in the case of the second cuboid conjecture, arXiv preprint, 2015.
Ruslan Sharipov, A note on invertible quadratic transformations of the real plane, arXiv preprint arXiv:1507.01861, 2015
Eric Weisstein's World of Mathematics, Euler Brick
Index entries for sequences related to bricks

Cf. A031173, A031175.

A031175 Shortest edge c of (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).

Original entry on oeis.org

44, 240, 140, 85, 160, 1008, 187, 429, 832, 780, 828, 1560, 528, 195, 1155, 1755, 495, 1575, 2964, 7840, 2925, 1008, 4368, 1080, 10296, 7579, 8789, 6072, 14112, 5643, 4599, 4900, 6435, 935, 7920, 7800, 4928, 7560, 23760, 1105, 2163, 2964

Offset: 1

Eric W. Weisstein

nonn

Primitive means that gcd(a,b,c) = 1.

See A031173 for a list of the 3356 primitive bricks with c < b < a < 5*10^8. - Giovanni Resta, Mar 23 2014

Calculated by F. Helenius (fredh(AT)ix.netcom.com).

Giovanni Resta, Table of n, a(n) for n = 1..3556
R. R. Gallyamov, I. R. Kadyrov, D. D. Kashelevskiy, A fast modulo primes algorithm for searching perfect cuboids and its implementation, arXiv preprint arXiv:1601.00636 [math.NT], 2016.
A. A. Masharov and R. A. Sharipov, A strategy of numeric search for perfect cuboids in the case of the second cuboid conjecture, arXiv preprint, 2015.
J. Ramsden and H. Sharipov, Inverse problems associated with perfect cuboids, arXiv preprint arXiv:1207.6764, 2012. - From _N. J. A. Sloane_, Dec 23 2012
J. Ramsden and H. Sharipov, On singularities of the inverse problems associated with perfect cuboids, arXiv preprint arXiv:1208.1859, 2012. - From _N. J. A. Sloane_, Dec 25 2012
Ruslan Sharipov, Perfect cuboids and irreducible polynomials, arXiv:1108.5348, 2011
Ruslan Sharipov, A note on the first cuboid conjecture, arXiv:1109.2534, 2011
Ruslan Sharipov, A note on the second cuboid conjecture. Part I, arXiv:1201.1229, 2012
Ruslan Sharipov, Perfect cuboids and multisymmetric polynomials, arXiv preprint arXiv:1205.3135, 2012. - From _N. J. A. Sloane_, Oct 22 2012
Ruslan Sharipov, On an ideal of multisymmetric polynomials associated with perfect cuboids, arXiv preprint arXiv:1206.6769, 2012. - From _N. J. A. Sloane_, Dec 17 2012
Ruslan Sharipov, On the equivalence of cuboid equations and their factor equations, arXiv preprint arXiv:1207.2102, 2012. - From _N. J. A. Sloane_, Dec 22 2012
Ruslan Sharipov, A biquadratic Diophantine equation associated with perfect cuboids, arXiv preprint arXiv:1207.4081, 2012. - From _N. J. A. Sloane_, Dec 23 2012
Ruslan Sharipov, On a pair of cubic equations associated with perfect cuboids, arXiv preprint arXiv:1208.0308, 2012. - From _N. J. A. Sloane_, Dec 23 2012
Ruslan Sharipov, On two elliptic curves associated with perfect cuboids, arXiv preprint arXiv:1208.1227, 2012. - From _N. J. A. Sloane_, Dec 24 2012
Ruslan Sharipov, Asymptotic estimates for roots of the cuboid characteristic equation in the linear region, arXiv preprint, 2015.
Ruslan Sharipov, Reverse asymptotic estimates for roots of the cuboid characteristic equation in the case of the second cuboid conjecture, arXiv preprint, 2015.
Ruslan Sharipov, A note on invertible quadratic transformations of the real plane, arXiv preprint arXiv:1507.01861, 2015
Eric Weisstein's World of Mathematics, Euler Brick
Index entries for sequences related to bricks

Cf. A031173, A031174.

A195816 Edge lengths of Euler bricks.

Original entry on oeis.org

44, 85, 88, 117, 132, 140, 160, 170, 176, 187, 195, 220, 231, 234, 240, 252, 255, 264, 275, 280, 308, 320, 340, 351, 352, 374, 390, 396, 420, 425, 429, 440, 462, 468, 480, 484, 495, 504, 510, 528, 550, 560, 561, 572, 585, 595, 616, 640, 660, 680, 693, 700

Offset: 1

Christopher Monckton of Brenchley, Oct 06 2011

nonn

Euler bricks are cuboids all of whose edges and face-diagonals are integers.

			For n=1, the edges (a,b,c) are (240,117,44) and the diagonals (d(a,b),d(a,c),d(b,c)) are (267,244,125).

L. E. Dickson, History of the Theory of Numbers, vol. 2, Diophantine Analysis, Dover, New York, 2005.
P. Halcke, Deliciae Mathematicae; oder, Mathematisches sinnen-confect., N. Sauer, Hamburg, Germany, 1719, page 265.

Robert G. Wilson v, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Euler brick

Cf. A031173, A031174, A031175.

ok[a_] := Catch[Block[{b, c, s}, s = Reduce[a^2 + b^2 == c^2 && b > 0 && c > 0, {b, c}, Integers]; If[s === False, Throw@ False, s = b /. List@ ToRules@ s]; Do[If[ IntegerQ@ Sqrt[s[[i]]^2 + s[[j]]^2], Throw@ True], {i, 2, Length@s}, {j, i - 1}]]; False]; Select[ Range[700], ok] (* Giovanni Resta, Nov 22 2018 *)

Integer edges a>b>c such that integer face-diagonals are d(a,b)=sqrt(a^2+b^2), d(a,c)=sqrt(a^2,c^2), d(b,c)=sqrt(b^2,c^2)

A118899 Surfaces of Euler bricks.

Original entry on oeis.org

87576, 334920, 350304, 391560, 693264, 788184, 993720, 1339680, 1401216, 1566240, 2189400, 2773056, 3014280, 3152736, 3524040, 3974880, 4205256, 4291224, 5358720, 5604864, 6239376, 6264960, 6881160, 7087080, 7093656, 8373000, 8757600, 8943480, 9789000, 10330080, 10596696, 11092224

Offset: 1

Giovanni Resta, May 05 2006

nonn

An Euler brick is a cuboid with integer sides and integer face diagonals.

			87576 is the surface of the Euler brick with sides 240, 117 and 44, whose face diagonals are 267, 244 and 125.

Robin Visser, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Euler Brick.

Cf. A031173, A118900, A118901, A118902.

More terms from Robin Visser, Jan 02 2024

A023185 Square of main diagonal of 3-dimensional box with coprime integer sides and integer face diagonals.

Original entry on oeis.org

73225, 196729, 543049, 706225, 730249, 3560089, 3584425, 6434041, 8392849, 14561209, 15686089, 40742425, 43508881, 69339625, 73878025, 85753369, 88450609, 90723169, 146947321, 148031689, 180998425, 216698161, 235198825, 273080809

Offset: 1

David W. Wilson

nonn

Ordered by length of main diagonal. - Sean A. Irvine, May 26 2019

Sean A. Irvine, Table of n, a(n) for n = 1..50
Sean A. Irvine, Java program (github)
Jean Lagrange, Sets of n Squares of Which Any n-1 Have Their Sum Square, Mathematics of Computation, Vol 41, Nr 164, Oct 1983.
Wikipedia, Euler brick

Cf. A031173, A031174, A031175.

A196943 Face-diagonal lengths of Euler bricks.

Original entry on oeis.org

125, 157, 244, 250, 267, 281, 314, 348, 365, 373, 375, 471, 488, 500, 534, 562, 625, 628, 696, 707, 725, 730, 732, 746, 750, 773, 785, 801, 808, 825, 843, 875, 942, 976, 979, 1000, 1037, 1044, 1068, 1095, 1099, 1119, 1124, 1125, 1193, 1220, 1250, 1256, 1335

Offset: 1

Christopher Monckton of Brenchley, Oct 07 2011

nonn

Euler bricks are cuboids all of whose edges and face-diagonals are integers.
It is not known whether any Euler brick with space-diagonals that are integers exists.
825 is the only integer common to the sets of edge lengths and of face-diagonal lengths <= 1000 for Euler bricks.

			For n=1, the edges (a,b,c) are (240,117,44) and the face-diagonals (d(a,b),d(a,c),d(b,c)) are (267,244,125).
Note the pleasing factorizations of the edge-lengths of this least Euler brick: 240 = 15*4^2; 117 = 13*3^2; 44 = 11*2^2.

L. E. Dickson, History of the Theory of Numbers, vol. 2, Diophantine Analysis, Dover, New York, 2005.
P. Halcke, Deliciae Mathematicae; oder, Mathematisches sinnen-confect., N. Sauer, Hamburg, Germany, 1719, page 265.

Robin Visser, Table of n, a(n) for n = 1..10000
E. W. Weisstein, MathWorld: Euler brick

cf. A195816, A031173, A031174, A031175. Edge lengths of Euler bricks are A195816.

Integer edges a > b > c such that integer face-diagonals are d(a,b) = sqrt(a^2 + b^2), d(a,c) = sqrt(a^2 + c^2), d(b,c) = sqrt(b^2 + c^2).

A118900 Volumes of Euler bricks.

Original entry on oeis.org

1235520, 8078400, 9884160, 16632000, 29272320, 33359040, 46569600, 64627200, 79073280, 133056000, 154440000, 218116800, 234178560, 266872320, 302132160, 372556800, 423783360, 449064000, 517017600, 632586240, 790352640, 883396800, 900694080, 924168960, 1009800000, 1064448000

Offset: 1

Giovanni Resta, May 05 2006

nonn

An Euler brick is a cuboid with integer sides and integer face diagonals.

			1235520 is the volume of the Euler brick with sides 240, 117 and 44, whose face diagonals are 267, 244 and 125.

Robin Visser, Table of n, a(n) for n = 1..10000 (terms 1..428 from Peter T. C. Radden).
Eric Weisstein's World of Mathematics, Euler Brick.

Cf. A031173, A118899, A118901, A118902.

Terms beyond a(21) from Peter T. C. Radden, Jan 12 2013

cp's OEIS Frontend

A298044 1/16 of the edge with the largest 2-adic valuation of a primitive 3-simplex (0, b=A031173, c=A031174, d=A031175).

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A298046 1/4 of the even edge of least 2-adic valuation of a primitive 3-simplex (0, b=A031173, c=A031174, d=A031175).

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A298047 The odd edge of a primitive 3-simplex (0, b=A031173, c=A031174, d=A031175).

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A031174 Intermediate edge b 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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A031175 Shortest edge c of (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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A195816 Edge lengths of Euler bricks.

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A118899 Surfaces of Euler bricks.

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A023185 Square of main diagonal of 3-dimensional box with coprime integer sides and integer face diagonals.

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A196943 Face-diagonal lengths of Euler bricks.

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A118900 Volumes of Euler bricks.

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