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A229839 Consider all 60-degree triangles with sides A < B < C. The sequence gives the values of C. 4
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 (list; graph; refs; listen; history; text; internal format)
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 A031103

Adjacent sequences:  A229836 A229837 A229838 * A229840 A229841 A229842

KEYWORD

nonn

AUTHOR

Colin Barker, Oct 01 2013

STATUS

approved

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Last modified December 6 15:18 EST 2021. Contains 349563 sequences. (Running on oeis4.)