A088977 Side of primitive equilateral triangle with prime cevian p=A002476(n) cutting an edge into two integral parts.

8, 15, 21, 35, 40, 48, 65, 77, 80, 91, 112, 117, 119, 133, 160, 168, 171, 187, 207, 209, 221, 224, 253, 255, 264, 280, 312, 323, 325, 341, 352, 377, 391, 403, 408, 425, 435, 440, 455, 465, 483, 504, 525, 527, 560, 576, 595, 609, 624, 645, 651, 665, 667, 703

Offset: 1

Author

Lekraj Beedassy, Oct 31 2003

Keywords

• nonn

Comments

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.

Links

• Jean-François Alcover, Table of n, a(n) for n = 1..2918
• F. Barnes, Deriving 60 degree triples

Crossrefs

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

Programs

Mathematica

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

Formula

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).

Extensions

More terms from Ray Chandler, Nov 01 2003