A088977 - OEIS

%I #20 Jul 07 2023 14:59:00

%S 8,15,21,35,40,48,65,77,80,91,112,117,119,133,160,168,171,187,207,209,

%T 221,224,253,255,264,280,312,323,325,341,352,377,391,403,408,425,435,

%U 440,455,465,483,504,525,527,560,576,595,609,624,645,651,665,667,703

%N Side of primitive equilateral triangle with prime cevian p=A002476(n) cutting an edge into two integral parts.

%C The edge a(n) is partitioned into q=s^2 - t^2=A088243(n)*A088296(n) and r=t(2s+t)=A088242(n)*A088299(n) by a cevian of length p. [Alternatively, (p,q,r) form a triangle with angle 2pi/3 opposite side p.] The quadruple {p,q,r,a(n)=q+r} satisfies the triangle relation: see A061281, or the simpler relation a(n)^2 = p^2 + q*r.

%H Jean-François Alcover, <a href="/A088977/b088977.txt">Table of n, a(n) for n = 1..2918</a>

%H F. Barnes, <a href="http://www.geocities.ws/fredlb37/node16.html">Deriving 60 degree triples</a>

%F a(n) = A088241(n)*A088298(n) = s(s+2t), where s^2 + st + t^2, with s>t, form the primes p = 1 (mod 6) = A002476(n).

%t sol[p_] := Solve[0 < t < s && s^2 + s t + t^2 == p, {s, t}, Integers];

%t Union[Reap[For[n = 1, n <= 10000, n++, If[PrimeQ[p = 6n + 1], an = s(s + 2t) /. sol[p][[1]]]; Sow[an]]][[2, 1]]] (* _Jean-François Alcover_, Mar 06 2020 *)

%Y Cf. A002476, A088241, A088242, A088243, A088296, A088298, A088299.

%K nonn

%O 1,1

%A _Lekraj Beedassy_, Oct 31 2003

%E More terms from _Ray Chandler_, Nov 01 2003