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

1,1

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

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).

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 *)

Crossrefs

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

Sequence in context: A274290 A328410 A089025 * A070043 A003786 A008686

Adjacent sequences: A088974 A088975 A088976 * A088978 A088979 A088980

Keyword

nonn

Author

Lekraj Beedassy, Oct 31 2003

Extensions

More terms from Ray Chandler, Nov 01 2003

Status

approved