A335895 Middle side of primitive triples, in nondecreasing order, for integer-sided triangles whose angles A < B < C are in arithmetic progression.

7, 7, 13, 13, 19, 19, 31, 31, 37, 37, 43, 43, 49, 49, 61, 61, 67, 67, 73, 73, 79, 79, 91, 91, 91, 91, 97, 97, 103, 103, 109, 109, 127, 127, 133, 133, 133, 133, 139, 139, 151, 151, 157, 157, 163, 163, 169, 169, 181, 181, 193, 193, 199, 199, 211, 211, 217, 217, 217, 217

Offset: 1

Author

Bernard Schott, Jul 04 2020

Keywords

• nonn

Comments

Equivalently, lengths of the middle side b of primitive non-equilateral triangles that have an angle of Pi/3; indeed, this side is opposite to angle B = Pi/3.

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

For the corresponding primitive triples and miscellaneous properties and references, see A335893.

When (a, b, c) is a triple or a solution, then (c-a, b, c) is another solution, so every b in the data is present an even number of times (see examples).

From Bernard Schott, Apr 02 2021: (Start)

Terms are primes of the form 6k+1, or products of primes of the form 6k+1. Three observations:

-> The lengths b are in A004611 \ {1} without repetition, 1 corresponds to the equilateral triangle (1, 1, 1).

-> Every term appears 2^k (k>0) times consecutively and the smallest term that appears 2^k times is precisely A121940(k); see examples.

-> The terms that appear precisely twice consecutively are in A133290. (End)

Examples

			b = 7 appears twice because A121940(1) = 7 and:
  7^2 = 3^2 - 3*8 + 8^2, with triple (3, 7, 8),
  7^2 = 5^2 - 5*8 + 8^2, with triple (5, 7, 8).
b = 91 appears four times because A121940(2) = 91 and:
  91^2 = 11^2 - 11*96 + 96^2, with triple (11, 91, 96),
  91^2 = 85^2 - 85*96 + 96^2, with triple (85, 91, 96),
  91^2 = 19^2 - 19*99 + 99^2, with triple (19, 91, 99),
  91^2 = 80^2 - 80*99 + 99^2, with triple (80, 91, 99).
b = 1729 appears eight times because A121940(3) = 1729 and the triples of these 2^3 = 8 triangles are (96, 1729, 1775), (1679, 1729, 1775), (249, 1729, 1840), (1591, 1729, 1840), (656, 1729, 1961), (1305, 1729, 1961), (799, 1729, 1984), (1185, 1729, 1984).
		

Crossrefs

Cf. A004611, A121940, A133290.

Cf. A335893 (triples), A335894 (smallest side), this sequence (middle side), A335896 (largest side), A335897 (perimeter).

Programs

Maple

for b from 3 to 100 by 2 do
for a from 1 to b-1 do
c := (a+ sqrt(4*b^2-3*a^2))/2;
if igcd(a, b) = 1 and issqr(4*b^2-3*a^2) then print(b); end if;
end do;
end do;

PARI

lista(nn) = {forstep(b=1, nn, 2, for(a=1, b-1, if (gcd(a, b) == 1, my(d = 4*b^2 - 3*a^2); if (issquare(d), my(c = (a + sqrtint(d))/2); if (denominator(c)==1, print1(b, ", "));););););} \\ Michel Marcus, Jul 05 2020

Formula

a(n) = A335893(n, 2).

b is such that b^2 = a^2 - a*c + c^2 with gcd(a,b) = 1 and a < b.