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

8, 15, 16, 21, 24, 30, 32, 35, 40, 42, 45, 48, 55, 56, 60, 63, 64, 65, 70, 72, 75, 77, 80, 84, 88, 90, 91, 96, 99, 104, 105, 110, 112, 117, 119, 120, 126, 128, 130, 133, 135, 136, 140, 143, 144, 147, 150, 152, 153, 154, 160, 165, 168, 171, 175, 176, 180, 182

Offset: 1

Author

Colin Barker, Oct 01 2013

Keywords

• nonn

Comments

A009005 gives the values of A, and A050931 gives the values of B.

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

Examples

			16 appears in the sequence because there exists a 60-degree triangle with sides 6, 14 and 16.
		

Links

• Wikipedia, Integer triangle
• Wikipedia, Cevian

Crossrefs

Cf. A009005, A050931, A229838.

Cf. A246918, A246919, A246920.

Programs

Mathematica

list={};cmax=182;
Do[If[IntegerQ[Sqrt[e^2-e t+t^2]],AppendTo[list,e]],{e,2,cmax},{t,1,e-1}]
list//DeleteDuplicates (* Herbert Kociemba, Apr 25 2021 *)

PARI

\\ Gives values of C not exceeding cmax.
\\ e.g. t60c(60) gives [8, 15, 16, 21, 24, 30, 32, 35, 40, 42, 45, 48, 55, 56, 60]
t60c(cmax) = {
  v=pt60c(cmax);
  s=[];
  for(i=1, #v,
    for(m=1, cmax\v[i],
      if(v[i]*m<=cmax, s=concat(s, v[i]*m))
    )
  );
  vecsort(s,,8)
}
\\ Gives values of C not exceeding cmax in primitive triangles.
\\ e.g. pt60c(115) gives [8, 15, 21, 35, 40, 48, 55, 65, 77, 80, 91, 96, 99, 112]
pt60c(cmax) = {
  s=[];
  for(m=1, ceil(sqrt(cmax+1)),
   for(n=1, m-1,
      if((m-n)%3!=0 && gcd(m, n)==1,
        if(2*m*n+m*m<=cmax, s=concat(s, 2*m*n+m*m))
      )
    )
  );
  vecsort(s,,8)
}