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

7, 13, 19, 31, 37, 43, 49, 61, 67, 73, 79, 91, 91, 97, 103, 109, 127, 133, 133, 139, 151, 157, 163, 169, 181, 193, 199, 211, 217, 217, 223, 229, 241, 247, 247, 259, 259, 271, 277, 283, 301, 301, 307, 313, 331, 337, 343, 349, 361, 367, 373, 379, 397, 403, 403, 409, 421, 427, 427, 433, 439, 457

Offset: 1

Author

Bernard Schott, Oct 01 2022

Keywords

• nonn

Comments

For the corresponding primitive triples and miscellaneous properties and references, see A357274.

Solutions c of the Diophantine equation c^2 = a^2 + a*b + b^2 with gcd(a,b) = 1 and a < b.

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.

Some properties:

-> Terms are primes of the form 6k+1, or products of primes of the form 6k+1.

-> The lengths c are in A004611 \ {1} without repetition, in increasing order.

-> Every term appears 2^(k-1) (k>=1) times consecutively.

-> The smallest term that appears 2^(k-1) times is precisely A121940(k): see examples.

-> The terms that appear only once in this sequence are in A133290.

-> 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).

Differs from A088513, the first 20 terms are the same then a(21) = 151 while A088513(21) = 157.

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.

Examples

			c = 7 appears once because A121940(1) = 7 with triple (3,5,7) and 7^2 = 3^2 + 3*5 + 5^2.
c = 91 is the smallest term to appear twice because A121940(2) = 91 with primitive 120-triples (11, 85, 91) and (19, 80, 91).
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).
		

Links

• Emrys Read, On integer-sided triangles containing angles of 120° or 60°, Mathematical Gazette 90, July 2006, 299-305.

Crossrefs

Cf. A357274 (triples), A357275(smallest side), A357276 (middle side), A357278 (perimeter).

Cf. Also A004611, A050931, A088513, A121940, A133290, A335895.

Programs

Maple

for c from 5 to 500 by 2 do
for a from 3 to c-2 do
b := (-a + sqrt(4*c^2-3*a^2))/2;
if b=floor(b) and gcd(a, b)=1 and a
				

Formula

a(n) = A357274(n, 3).