A070137 - OEIS

%I #20 Oct 04 2021 08:40:25

%S 17,212,493,1297,2574,4298,5251,14414,16365,21231,26125,39056,42597,

%T 55042,63770,75052,91121,97256,124355,164640,200999,213083,253721,

%U 275999,367997,384154,415778,478343,511633,518370,606417,665040,689356,755435,846571

%N Numbers k such that [A070080(k), A070081(k), A070082(k)] is a right integer triangle with relatively prime side lengths.

%C Right integer triangles have integer areas: see A070143.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HeronianTriangle.html">Heronian Triangle</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RightTriangle.html">Right Triangle</a>.

%H Reinhard Zumkeller, <a href="/A070080/a070080.txt">Integer-sided triangles</a>

%e 493 is a term: [A070080(493), A070081(493), A070082(493)]=[8,15,17], A070084(493)=gcd(8,15,17)=1, A070085(493)=8^2+15^2-17^2=64+225-289=0.

%t m = 500 (* max perimeter *);

%t sides[per_] := Select[Reverse /@ IntegerPartitions[per, {3}, Range[ Ceiling[per/2]]], #[[1]] < per/2 && #[[2]] < per/2 && #[[3]] < per/2 &];

%t triangles = DeleteCases[Table[sides[per], {per, 3, m}], {}] // Flatten[#, 1] & // SortBy[Total[#] m^3 + #[[1]] m^2 + #[[2]] m + #[[1]] &];

%t Position[triangles, {a_, b_, c_} /; GCD[a, b, c] == 1 && a^2 + b^2 - c^2 == 0] // Flatten (* _Jean-François Alcover_, Oct 04 2021 *)

%Y Cf. A070109, A070136.

%K nonn

%O 1,1

%A _Reinhard Zumkeller_, May 05 2002

%E More terms from _Jean-François Alcover_, Oct 04 2021