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A197040 Occurrences of edge-lengths of Euler bricks in every 100 consecutive integers.

%I A197040 #19 Feb 16 2025 08:33:15
%S A197040 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 A197040 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 A197040 7,6,9,9,6,8,7,9,8,6,9,5,9,9,8,6,6,7,7
%N A197040 Occurrences of edge-lengths of Euler bricks in every 100 consecutive integers.
%C A197040 Distribution of edge-length occurrences for Euler bricks is remarkably near-uniform.
%D A197040 L. E. Dickson, History of the Theory of Numbers, vol. 2, Diophantine Analysis, Dover, New York, 2005.
%D A197040 P. Halcke, Deliciae Mathematicae; oder, Mathematisches sinnen-confect., N. Sauer, Hamburg, Germany, 1719, page 265.
%H A197040 Robin Visser, <a href="/A197040/b197040.txt">Table of n, a(n) for n = 1..10000</a>
%H A197040 E. W. Weisstein, <a href="https://mathworld.wolfram.com/EulerBrick.html">MathWorld: Euler brick</a>
%e A197040 For n=1 (i.e., the integers 1..100), there are only 3 possible edge-lengths for Euler bricks: 44, 85, 88.
%o A197040 (Sage)
%o A197040 def a(n):
%o A197040     ans = set()
%o A197040     for x in range(100*(n-1)+1, 100*n+1):
%o A197040         divs = Integer(x^2).divisors()
%o A197040         for d in divs:
%o A197040             if (d <= x^2/d): continue
%o A197040             if (d-x^2/d)%2==0:
%o A197040                 y = (d-x^2/d)/2
%o A197040                 for e in divs:
%o A197040                     if (e <= x^2/e): continue
%o A197040                     if (e-x^2/e)%2==0:
%o A197040                         z = (e-x^2/e)/2
%o A197040                         if (y^2+z^2).is_square(): ans.add(x)
%o A197040     return len(ans)  # _Robin Visser_, Jan 02 2024
%Y A197040 cf. A195816, A196943, A031173, A031174, A031175. Edge lengths of Euler bricks are A195816; face diagonals are A196943.
%K A197040 nonn,base
%O A197040 1,1
%A A197040 _Christopher Monckton of Brenchley_, Oct 08 2011