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A084647 Hypotenuses for which there exist exactly 3 distinct integer triangles.

%I A084647 #37 Feb 16 2025 08:32:49
%S A084647 125,250,375,500,750,875,1000,1125,1375,1500,1750,2000,2197,2250,2375,
%T A084647 2625,2750,2875,3000,3375,3500,3875,4000,4125,4394,4500,4750,4913,
%U A084647 5250,5375,5500,5750,5875,6000,6125,6591,6750,7000,7125,7375,7750
%N A084647 Hypotenuses for which there exist exactly 3 distinct integer triangles.
%C A084647 Numbers whose square is decomposable in 3 different ways into the sum of two nonzero squares: these are those with exactly one prime divisor of the form 4k+1 with multiplicity three. - _Jean-Christophe Hervé_, Nov 11 2013
%H A084647 Ray Chandler, <a href="/A084647/b084647.txt">Table of n, a(n) for n = 1..10000</a> (first 1140 terms from Jean-Christophe Hervé)
%H A084647 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PythagoreanTriple.html">Pythagorean Triple</a>
%F A084647 Terms are obtained by the products A004144(k)*A002144(p)^3 for k, p > 0, ordered by increasing values. - _Jean-Christophe Hervé_, Nov 12 2013
%e A084647 a(1) = 125 = 5^3, and 125^2 = 100^2 + 75^2 = 117^2 + 44^2 = 120^2 + 35^2. - _Jean-Christophe Hervé_, Nov 11 2013
%t A084647 Clear[lst,f,n,i,k] f[n_]:=Module[{i=0,k=0},Do[If[Sqrt[n^2-i^2]==IntegerPart[Sqrt[n^2-i^2]],k++ ],{i,n-1,1,-1}]; k/2]; lst={}; Do[If[f[n]==3,AppendTo[lst,n]],{n,4*5!}]; lst (* _Vladimir Joseph Stephan Orlovsky_, Aug 12 2009 *)
%Y A084647 Cf. A002144, A006339, A046080, A046109, A083025.
%Y A084647 Cf. A004144 (0), A084645 (1), A084646 (2), A084648 (4), A084649 (5), A097219 (6), A097101 (7), A290499 (8), A290500 (9), A097225 (10), A290501 (11), A097226 (12), A097102 (13), A290502 (14), A290503 (15), A097238 (16), A097239 (17), A290504 (18), A290505 (19), A097103 (22), A097244 (31), A097245 (37), A097282 (40), A097626 (67).
%K A084647 nonn
%O A084647 1,1
%A A084647 _Eric W. Weisstein_, Jun 01 2003