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 A088977 Side of primitive equilateral triangle with prime cevian p=A002476(n) cutting an edge into two integral parts. 9
 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 (list; graph; refs; listen; history; text; internal format)
 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. REFERENCES Mohammad K. Azarian, A Trigonometric Characterization of  Equilateral Triangle, Problem 336, Mathematics and Computer Education, Vol. 31, No. 1, Winter 1997, p. 96.  Solution published in Vol. 32, No. 1, Winter 1998, pp. 84-85. Mohammad K. Azarian, Equating Distances and Altitude in an Equilateral Triangle, Problem 316, Mathematics and Computer Education, Vol. 28, No. 3, Fall 1994, p. 337.  Solution published in Vol. 29, No. 3, Fall 1995, pp. 324-325. 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][]]; 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

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Last modified September 18 01:39 EDT 2021. Contains 347504 sequences. (Running on oeis4.)