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A089025 Side of primitive equilateral triangle bearing at least one integral cevian that partitions an edge into two integral sections.

%I A089025 #38 Jul 07 2023 14:59:39
%S A089025 8,15,21,35,40,48,55,65,77,80,91,96,99,112,117,119,133,143,153,160,
%T A089025 168,171,176,187,207,209,221,224,225,247,253,255,264,275,280,285,299,
%U A089025 312,319,323,325,341,345,352,360,377,391,403,408,416,425,435,437,440,448
%N A089025 Side of primitive equilateral triangle bearing at least one integral cevian that partitions an edge into two integral sections.
%C A089025 The cevians are numbers divisible only by primes of form 6n+1:A002476 (i.e., correspond to entries of A004611).
%C A089025 Composite cevians c belong to more than one equilateral triangle, actually to 2^(omega(c)-1) of them, where omega(n)=A001221(n). For instance, cevian 1813=7^2*37, with omega(1813)=2, belongs to 2^(2-1)=2 equilateral triangles, their sides being 1927=255+1627 and 1960=343+1617, while cevian 1729=7*13*19, with omega(1729)=3, belongs to 2^(3-1)=4 equilateral triangles whose sides are 1775=96+1679, 1824=209+1615, 1840=249+1591, 1859=299+1560.
%C A089025 Given a triangle with integer side lengths a, b, c relatively prime with a < b, c < b, and angle opposite c of 60 degrees then a*a - a*b + b*b = c*c from law of cosines and called a primitive Eisenstein triple by Gordon. This sequence is the possible side lengths of b. - _Michael Somos_, Apr 11 2012
%H A089025 O. Delgado-Friedrichs and M. O'Keeffe, <a href="http://dx.doi.org/10.1107/S0108767309026270">Edge-transitive lattice nets</a>, Acta Cryst. A, A65 (2009), 360-363.
%H A089025 Russell A. Gordon, <a href="http://www.jstor.org/stable/10.4169/math.mag.85.1.12">Properties of Eisenstein Triples</a>, Mathematics Magazine 85 (2012), 12-25.
%e A089025 The equilateral triangle with side 280, for instance, has cevian 247 partitioning an edge into 93+187, as well as cevian 271 that sections the edge into 19+261.
%t A089025 findPrimIntEquiSide[maxC_] :=
%t A089025 Reap[Do[Do[
%t A089025      With[{cevian = Abs[c E^((2 \[Pi] I)/6) - a]},
%t A089025       If[FractionalPart[cevian] == 0 && GCD[a, c] == 1,
%t A089025        Sow[c]; Break[]]], {a, Floor[c/2],
%t A089025       1, -1}], {c, maxC}]][[2, 1]]
%t A089025 (* _Andrew Turner_, Aug 04 2017 *)
%Y A089025 Cf. A088513, A088514, A088977.
%K A089025 nonn
%O A089025 1,1
%A A089025 _Lekraj Beedassy_, Nov 12 2003