|
%I #14 Oct 04 2021 08:04:25
%S 1,2,4,5,6,7,8,11,12,13,14,15,16,17,19,20,21,22,23,25,27,28,29,30,32,
%T 33,35,36,37,39,40,41,42,43,44,45,46,47,49,51,52,53,55,56,57,58,59,60,
%U 61,62,63,64,65,66,67,69,70,72,73,74,75,77
%N Numbers k such that [A070080(k), A070081(k), A070082(k)] is an integer triangle with relatively prime side lengths.
%C A070084(a(k)) = gcd(A070080(a(k)), A070081(a(k)), A070082(a(k))) = 1;
%C all integer triangles [A070080(a(k)), A070081(a(k)), A070082(a(k))] are mutually nonisomorphic.
%H Jean-François Alcover, <a href="/A070110/b070110.txt">Table of n, a(n) for n = 1..789</a>
%H Reinhard Zumkeller, <a href="/A070080/a070080.txt">Integer-sided triangles</a>
%e 13 is a term: [A070080(13), A070081(13), A070082(13)]=[2,4,5], A070084(13)=gcd(2,4,5)=1.
%t m = 50 (* 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] // Flatten (* _Jean-François Alcover_, Oct 04 2021 *)
%Y Cf. A051493, A070113, A070116, A070119, A070122, A070125, A070128, A070131, A070134, A070137, A070084.
%K nonn
%O 1,2
%A _Reinhard Zumkeller_, May 05 2002
|
