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A070122 Numbers m such that [A070080(m), A070081(m), A070082(m)] is an acute scalene integer triangle with relatively prime side lengths.

%I A070122 #12 Oct 12 2021 21:57:08
%S A070122 33,45,53,60,70,83,90,92,106,114,119,132,134,142,148,162,165,168,175,
%T A070122 181,183,197,200,203,204,218,224,237,240,245,247,261,264,267,268,282,
%U A070122 290,293,296,309,316,317,319,333,341,345,348
%N A070122 Numbers m such that [A070080(m), A070081(m), A070082(m)] is an acute scalene integer triangle with relatively prime side lengths.
%H A070122 Jean-François Alcover, <a href="/A070122/b070122.txt">Table of n, a(n) for n = 1..229</a>
%H A070122 Reinhard Zumkeller, <a href="/A070080/a070080.txt">Integer-sided triangles</a>
%e A070122 70 is a term because [A070080(70), A070081(70), A070082(70)]=[5<7<8], A070084(70)=gcd(5,7,8)=1, A070085(70)=5^2+7^2-8^2=25+49-64=10>0.
%t A070122 m = 55 (* max perimeter *);
%t A070122 sides[per_] := Select[Reverse /@ IntegerPartitions[per, {3}, Range[ Ceiling[per/2]]], #[[1]] < per/2 && #[[2]] < per/2 && #[[3]] < per/2 &];
%t A070122 triangles = DeleteCases[Table[sides[per], {per, 3, m}], {}] // Flatten[#, 1]& // SortBy[Total[#] m^3 + #[[1]] m^2 + #[[2]] m + #[[1]]&];
%t A070122 Position[triangles, {a_, b_, c_} /; a < b < c && GCD[a, b, c] == 1 && a^2 + b^2 - c^2 > 0] // Flatten (* _Jean-François Alcover_, Oct 12 2021 *)
%Y A070122 Cf. A070096, A070118, A070110, A070112.
%K A070122 nonn
%O A070122 1,1
%A A070122 _Reinhard Zumkeller_, May 05 2002