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A084648 Hypotenuses for which there exist exactly 4 distinct integer triangles.

%I A084648 #40 Feb 16 2025 08:32:49
%S A084648 65,85,130,145,170,185,195,205,221,255,260,265,290,305,340,365,370,
%T A084648 377,390,410,435,442,445,455,481,485,493,505,510,520,530,533,545,555,
%U A084648 565,580,585,595,610,615,625,629,663,680,685,689,697,715,730,740,745
%N A084648 Hypotenuses for which there exist exactly 4 distinct integer triangles.
%C A084648 Numbers whose square is decomposable in 4 different ways into the sum of two nonzero squares: these are those with exactly 2 distinct prime divisors of the form 4k+1, each with multiplicity one, or with only one prime divisor of this form with multiplicity 4. - _Jean-Christophe Hervé_, Nov 11 2013
%C A084648 If m is a term, then 2*m and p*m are terms where p is any prime of the form 4k+3. - _Ray Chandler_, Dec 30 2019
%H A084648 Ray Chandler, <a href="/A084648/b084648.txt">Table of n, a(n) for n = 1..10000</a>
%H A084648 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PythagoreanTriple.html">Pythagorean Triple</a>
%e A084648 a(1) = 65 = 5*13, and 65^2 = 52^2 + 39^2 = 56^2 + 33^2 = 60^2 + 25^2 = 63^2 + 16^2. - _Jean-Christophe Hervé_, Nov 11 2013
%t A084648 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]==4,AppendTo[lst,n]],{n,6!}]; lst (* _Vladimir Joseph Stephan Orlovsky_, Aug 12 2009 *)
%Y A084648 Cf. A002144, A006339, A046080, A046109, A083025.
%Y A084648 Cf. A004144 (0), A084645 (1), A084646 (2), A084647 (3), 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 A084648 nonn
%O A084648 1,1
%A A084648 _Eric W. Weisstein_, Jun 01 2003