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

8, 15, 21, 35, 40, 48, 55, 65, 77, 80, 91, 96, 99, 112, 117, 119, 133, 143, 153, 160, 168, 171, 176, 187, 207, 209, 221, 224, 225, 247, 253, 255, 264, 275, 280, 285, 299, 312, 319, 323, 325, 341, 345, 352, 360, 377, 391, 403, 408, 416, 425, 435, 437, 440, 448

Offset

1,1

Comments

The cevians are numbers divisible only by primes of form 6n+1:A002476 (i.e., correspond to entries of A004611).

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.

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

Links

Table of n, a(n) for n=1..55.

O. Delgado-Friedrichs and M. O'Keeffe, Edge-transitive lattice nets, Acta Cryst. A, A65 (2009), 360-363.

Russell A. Gordon, Properties of Eisenstein Triples, Mathematics Magazine 85 (2012), 12-25.

Example

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.

Mathematica

findPrimIntEquiSide[maxC_] :=
Reap[Do[Do[
     With[{cevian = Abs[c E^((2 \[Pi] I)/6) - a]},
      If[FractionalPart[cevian] == 0 && GCD[a, c] == 1,
       Sow[c]; Break[]]], {a, Floor[c/2],
      1, -1}], {c, maxC}]][[2, 1]]
(* Andrew Turner, Aug 04 2017 *)

Crossrefs

Cf. A088513, A088514, A088977.

Sequence in context: A374476 A274290 A328410 * A088977 A070043 A003786

Adjacent sequences: A089022 A089023 A089024 * A089026 A089027 A089028

Keyword

nonn

Author

Lekraj Beedassy, Nov 12 2003

Status

approved