%I #19 Feb 16 2025 08:33:15

%S 3,8,9,8,9,9,6,9,10,8,7,9,6,8,7,8,11,6,7,8,9,8,7,6,8,10,6,6,6,8,8,8,8,

%T 9,6,9,7,6,7,8,8,9,7,11,7,8,5,9,8,9,9,7,6,7,9,6,7,9,7,8,10,5,9,7,7,7,

%U 7,6,9,9,6,8,7,9,8,6,9,5,9,9,8,6,6,7,7

%N Occurrences of edge-lengths of Euler bricks in every 100 consecutive integers.

%C Distribution of edge-length occurrences for Euler bricks is remarkably near-uniform.

%D L. E. Dickson, History of the Theory of Numbers, vol. 2, Diophantine Analysis, Dover, New York, 2005.

%D P. Halcke, Deliciae Mathematicae; oder, Mathematisches sinnen-confect., N. Sauer, Hamburg, Germany, 1719, page 265.

%H Robin Visser, <a href="/A197040/b197040.txt">Table of n, a(n) for n = 1..10000</a>

%H E. W. Weisstein, <a href="https://mathworld.wolfram.com/EulerBrick.html">MathWorld: Euler brick</a>

%e For n=1 (i.e., the integers 1..100), there are only 3 possible edge-lengths for Euler bricks: 44, 85, 88.

%o (Sage)

%o def a(n):

%o ans = set()

%o for x in range(100*(n-1)+1, 100*n+1):

%o divs = Integer(x^2).divisors()

%o for d in divs:

%o if (d <= x^2/d): continue

%o if (d-x^2/d)%2==0:

%o y = (d-x^2/d)/2

%o for e in divs:

%o if (e <= x^2/e): continue

%o if (e-x^2/e)%2==0:

%o z = (e-x^2/e)/2

%o if (y^2+z^2).is_square(): ans.add(x)

%o return len(ans) # _Robin Visser_, Jan 02 2024

%Y cf. A195816, A196943, A031173, A031174, A031175. Edge lengths of Euler bricks are A195816; face diagonals are A196943.

%K nonn,base

%O 1,1

%A _Christopher Monckton of Brenchley_, Oct 08 2011