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A357277 Largest side c of primitive triples, in nondecreasing order, for integer-sided triangles with angles A < B < C = 2*Pi/3 = 120 degrees.

%I A357277 #27 Jan 29 2023 19:45:54
%S A357277 7,13,19,31,37,43,49,61,67,73,79,91,91,97,103,109,127,133,133,139,151,
%T A357277 157,163,169,181,193,199,211,217,217,223,229,241,247,247,259,259,271,
%U A357277 277,283,301,301,307,313,331,337,343,349,361,367,373,379,397,403,403,409,421,427,427,433,439,457
%N A357277 Largest side c of primitive triples, in nondecreasing order, for integer-sided triangles with angles A < B < C = 2*Pi/3 = 120 degrees.
%C A357277 For the corresponding primitive triples and miscellaneous properties and references, see A357274.
%C A357277 Solutions c of the Diophantine equation c^2 = a^2 + a*b + b^2 with gcd(a,b) = 1 and a < b.
%C A357277 Also, side c can be generated with integers u, v such that gcd(u,v) = 1 and 0 < v < u, c = u^2 + u*v + v^2.
%C A357277 Some properties:
%C A357277 -> Terms are primes of the form 6k+1, or products of primes of the form 6k+1.
%C A357277 -> The lengths c are in A004611 \ {1} without repetition, in increasing order.
%C A357277 -> Every term appears 2^(k-1) (k>=1) times consecutively.
%C A357277 -> The smallest term that appears 2^(k-1) times is precisely A121940(k): see examples.
%C A357277 -> The terms that appear only once in this sequence are in A133290.
%C A357277 -> The terms are the same as in A335895 but frequency is not the same: when a term appears m times consecutively here, it appears 2m times consecutively in A335895. This is because if (a,b,c) is a primitive 120-triple, then both (a,a+b,c) and (a+b,b,c) are 60-triples in A335893 (see Emrys Read link, lemma 2 p. 302).
%C A357277 Differs from A088513, the first 20 terms are the same then a(21) = 151 while A088513(21) = 157.
%C A357277 A050931 gives all the possible values of the largest side c, in increasing order without repetition, for all triangles with an angle of 120 degrees, but not necessarily primitive.
%H A357277 Emrys Read, <a href="https://doi.org/10.1017/S0025557200179793">On integer-sided triangles containing angles of 120° or 60°</a>, Mathematical Gazette 90, July 2006, 299-305.
%F A357277 a(n) = A357274(n, 3).
%e A357277 c = 7 appears once because A121940(1) = 7 with triple (3,5,7) and 7^2 = 3^2 + 3*5 + 5^2.
%e A357277 c = 91 is the smallest term to appear twice because A121940(2) = 91 with primitive 120-triples (11, 85, 91) and (19, 80, 91).
%e A357277 c = 1729 is the smallest term to appear four times because A121940(3) = 1729 with triples (96, 1679, 1729), (249, 1591, 1729), (656, 1305, 1729), (799, 1185, 1729).
%p A357277 for c from 5 to 500 by 2 do
%p A357277 for a from 3 to c-2 do
%p A357277 b := (-a + sqrt(4*c^2-3*a^2))/2;
%p A357277 if b=floor(b) and gcd(a, b)=1 and a<b then print(c); end if;
%p A357277 end do;
%p A357277 end do;
%Y A357277 Cf. A357274 (triples), A357275(smallest side), A357276 (middle side), A357278 (perimeter).
%Y A357277 Cf. Also A004611, A050931, A088513, A121940, A133290, A335895.
%K A357277 nonn
%O A357277 1,1
%A A357277 _Bernard Schott_, Oct 01 2022