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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
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
Sequence in context: A248389 A301618 A192915 * A114605 A300860 A352989
KEYWORD
nonn
AUTHOR
Colin Barker, Oct 01 2013
STATUS
approved

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Last modified May 27 11:20 EDT 2024. Contains 372858 sequences. (Running on oeis4.)