A195816 Edge lengths of Euler bricks.

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

1,1

Comments

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

References

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.

Links

Robert G. Wilson v, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Euler brick

Formula

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)

Example

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).

Mathematica

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 *)

Crossrefs

Cf. A031173, A031174, A031175.

Sequence in context: A050272 A260201 A112721 * A118087 A335481 A345409

Adjacent sequences: A195813 A195814 A195815 * A195817 A195818 A195819

Keyword

nonn

Author

Christopher Monckton of Brenchley, Oct 06 2011

Status

approved