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

1,1

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

Links

Table of n, a(n) for n=1..58.

Wikipedia, Integer triangle

Wikipedia, Cevian

Example

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

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 *)

Prog

(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)
}

Crossrefs

Cf. A009005, A050931, A229838.

Cf. A246918, A246919, A246920.

Sequence in context: A248389 A301618 A192915 * A114605 A300860 A352989

Adjacent sequences: A229836 A229837 A229838 * A229840 A229841 A229842

Keyword

nonn

Author

Colin Barker, Oct 01 2013

Status

approved