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A229839 Consider all 60-degree triangles with sides A < B < C. The sequence gives the values of C.

%I A229839 #22 May 02 2021 21:52:15
%S A229839 8,15,16,21,24,30,32,35,40,42,45,48,55,56,60,63,64,65,70,72,75,77,80,
%T A229839 84,88,90,91,96,99,104,105,110,112,117,119,120,126,128,130,133,135,
%U A229839 136,140,143,144,147,150,152,153,154,160,165,168,171,175,176,180,182
%N A229839 Consider all 60-degree triangles with sides A < B < C. The sequence gives the values of C.
%C A229839 A009005 gives the values of A, and A050931 gives the values of B.
%C A229839 The side n of an equilateral triangle for which a nontrivial integral cevian of length less than n exists, which divides an edge into two integral parts. - _Colin Barker_, Sep 09 2014
%H A229839 Wikipedia, <a href="https://en.wikipedia.org/wiki/Integer_triangle">Integer triangle</a>
%H A229839 Wikipedia, <a href="http://en.wikipedia.org/wiki/Cevian">Cevian</a>
%e A229839 16 appears in the sequence because there exists a 60-degree triangle with sides 6, 14 and 16.
%t A229839 list={};cmax=182;
%t A229839 Do[If[IntegerQ[Sqrt[e^2-e t+t^2]],AppendTo[list,e]],{e,2,cmax},{t,1,e-1}]
%t A229839 list//DeleteDuplicates (* _Herbert Kociemba_, Apr 25 2021 *)
%o A229839 (PARI)
%o A229839 \\ Gives values of C not exceeding cmax.
%o A229839 \\ e.g. t60c(60) gives [8, 15, 16, 21, 24, 30, 32, 35, 40, 42, 45, 48, 55, 56, 60]
%o A229839 t60c(cmax) = {
%o A229839   v=pt60c(cmax);
%o A229839   s=[];
%o A229839   for(i=1, #v,
%o A229839     for(m=1, cmax\v[i],
%o A229839       if(v[i]*m<=cmax, s=concat(s, v[i]*m))
%o A229839     )
%o A229839   );
%o A229839   vecsort(s,,8)
%o A229839 }
%o A229839 \\ Gives values of C not exceeding cmax in primitive triangles.
%o A229839 \\ e.g. pt60c(115) gives [8, 15, 21, 35, 40, 48, 55, 65, 77, 80, 91, 96, 99, 112]
%o A229839 pt60c(cmax) = {
%o A229839   s=[];
%o A229839   for(m=1, ceil(sqrt(cmax+1)),
%o A229839    for(n=1, m-1,
%o A229839       if((m-n)%3!=0 && gcd(m, n)==1,
%o A229839         if(2*m*n+m*m<=cmax, s=concat(s, 2*m*n+m*m))
%o A229839       )
%o A229839     )
%o A229839   );
%o A229839   vecsort(s,,8)
%o A229839 }
%Y A229839 Cf. A009005, A050931, A229838.
%Y A229839 Cf. A246918, A246919, A246920.
%K A229839 nonn
%O A229839 1,1
%A A229839 _Colin Barker_, Oct 01 2013