A031175 -id:A031175 - 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.

A031173 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).

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein

nonn

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

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 arXiv:1504.07161 [math.NT], 2015.
Dustin Moody, Mohammad Sadek, Arman Shamsi Zargar, Families of elliptic curves of rank >= 5 over Q(t), Rocky Mountain Journal of Mathematics (2019) Vol. 49, No. 7, 2253-2266.
Giovanni Resta, The 3556 primitive bricks with c < b < a < 5*10^8
J. Ramsden and H. Sharipov, Inverse problems associated with perfect cuboids, arXiv preprint arXiv:1207.6764 [math.NT], 2012.
J. Ramsden and H. Sharipov, On singularities of the inverse problems associated with perfect cuboids, arXiv preprint arXiv:1208.1859 [math.NT], 2012.
Ruslan Sharipov, Perfect cuboids and irreducible polynomials, arXiv:1108.5348 [math.NT], 2011.
Ruslan Sharipov, A note on the first cuboid conjecture, arXiv:1109.2534 [math.NT], 2011.
Ruslan Sharipov, A note on the second cuboid conjecture. Part I, arXiv:1201.1229 [math.NT], 2012.
Ruslan Sharipov, Perfect cuboids and multisymmetric polynomials, arXiv preprint arXiv:1205.3135 [math.NT], 2012.
Ruslan Sharipov, On an ideal of multisymmetric polynomials associated with perfect cuboids, arXiv preprint arXiv:1206.6769 [math.NT], 2012.
Ruslan Sharipov, On the equivalence of cuboid equations and their factor equations, arXiv preprint arXiv:1207.2102 [math.NT], 2012.
Ruslan Sharipov, A biquadratic Diophantine equation associated with perfect cuboids, arXiv preprint arXiv:1207.4081 [math.NT], 2012.
Ruslan Sharipov, On a pair of cubic equations associated with perfect cuboids, arXiv preprint arXiv:1208.0308 [math.NT], 2012.
Ruslan Sharipov, On two elliptic curves associated with perfect cuboids, arXiv preprint arXiv:1208.1227 [math.NT], 2012.
Ruslan Sharipov, Asymptotic estimates for roots of the cuboid characteristic equation in the linear region, arXiv preprint arXiv:1505.02745 [math.NT], 2015.
Ruslan Sharipov, Reverse asymptotic estimates for roots of the cuboid characteristic equation in the case of the second cuboid conjecture, arXiv preprint arXiv:1505.00724 [math.NT], 2015.
Ruslan Sharipov, A note on invertible quadratic transformations of the real plane, arXiv preprint arXiv:1507.01861 [math.AG], 2015.
Eric Weisstein's World of Mathematics, Euler Brick
Walter Wyss, No Perfect Cuboid, arXiv:1506.02215 [math.NT], 2015-2017.
Index entries for sequences related to bricks

Cf. 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.

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)

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).

A268396 Sides of Pythagorean cuboids: triples (a, b, c) that are integral length sides of a rectangular cuboid for which the three face diagonals x, y, z also have integral length.

Original entry on oeis.org

44, 117, 240, 240, 252, 275, 88, 234, 480, 85, 132, 720, 160, 231, 792, 132, 351, 720, 140, 480, 693, 480, 504, 550, 176, 468, 960, 170, 264, 1440, 220, 585, 1200, 720, 756, 825, 320, 462, 1584, 264, 702, 1440, 280, 960, 1386, 187, 1020, 1584, 308, 819, 1680

Offset: 1

Arkadiusz Wesolowski, Feb 03 2016

nonn

Sides in increasing order of perimeter (a+b+c), where a < b < c.
A triple (a, b, c) of integers belongs to this sequence if and only if all of the numbers sqrt(a^2 + b^2), sqrt(b^2 + c^2), and sqrt(a^2 + c^2) are also integers.
Consider the set S(n) = {a(3*n-2), a(3*n-1), a(3*n)}. Then:
- at least one number in the set is divisible by 5
- at least one number in the set is divisible by 9
- at least one number in the set is divisible by 11
- at least one number in the set is divisible by 16
- at least two numbers in the set are divisible by 3
- at least two numbers in the set are divisible by 4.
The list of "Sides of ..." is A195816, while this sequence lists "Triples...", i.e., (a(3n-2), a(3n-1), a(3n)) = (A031175(k), A031174(k), A031173(k)) for some k, n >= 1. (The order is not the same as for A031173 etc, e.g., the 5th through 8th triple have decreasing largest sides.) Also, in A031173, A031174, A031175 and others, the side naming convention is a > b > c, the opposite of here. - M. F. Hasler, Oct 11 2018

Eli Maor, The Pythagorean Theorem: A 4,000-Year History, 2007, Princeton University Press, p. 134.

Wikipedia, Cuboid
Wikipedia, Euler brick

Cf. A195816.
See A245616 for a very similar sequence.
Cf. A031173, A031174, A031175.

A295507 1/15840 of the volume of a primitive 3-simplex.

Original entry on oeis.org

13, 175, 490, 85, 308, 13475, 3179, 9295, 19760, 60775, 55315, 225420, 203665, 9724, 512050, 568854, 207740, 252700, 2684396, 8695050, 1221025, 140875, 3121937, 311220, 21390460, 12289004, 17331908, 20074032, 44144100, 18055947, 21118608, 21017325, 40245205, 4418232

Offset: 1

Ralf Steiner, Nov 23 2017

nonn

The volume of every primitive trirectangular tetrahedron with integer sides is divisible by 15840. See A031173 for more comments.

These primitive trirectangular tetrahedrons are primitive 3-simplices.

Eric Weisstein's World of Mathematics, Trirectangular Tetrahedron

Cf. A031173, A031174, A031175.

a(n) = (1/15840)*A031173(n)*A031174(n)*A031175(n)/6.

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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A031173 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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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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A195816 Edge lengths 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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A268396 Sides of Pythagorean cuboids: triples (a, b, c) that are integral length sides of a rectangular cuboid for which the three face diagonals x, y, z also have integral length.

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A295507 1/15840 of the volume of a primitive 3-simplex.

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