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A350038 Numbers that are the perimeter of a primitive 60-degree integer triangle.

%I A350038 #27 Dec 11 2021 04:29:21
%S A350038 18,20,35,36,45,56,77,84,90,104,110,120,126,135,143,170,176,182,189,
%T A350038 198,209,210,216,221,252,260,264,266,270,272,273,297,299,323,350,351,
%U A350038 360,368,374,378,380,390,396,425,432,437,459,462,464,468,476,494,495,506,527,551,561,570,575,585,594,608,612
%N A350038 Numbers that are the perimeter of a primitive 60-degree integer triangle.
%H A350038 Seiichi Manyama, <a href="/A350038/b350038.txt">Table of n, a(n) for n = 1..10000</a>
%e A350038 b(n) = Sum_{k=1..3} A264826(3*n+k-3).
%e A350038 c(n) = Sum_{k=1..3} A201223(3*n+k-3).
%e A350038 b(1) = c(1) = 3+7+8 = 18 = a(1).
%e A350038 b(2) = c(2) = 5+7+8 = 20 = a(2).
%e A350038 b(3) = c(5) = 5+19+21 = 45 = a(5).
%e A350038 b(4) = c(3) = 7+13+15 = 35 = a(3).
%e A350038 b(5) = c(9) = 7+37+40 = 84 = a(8).
%e A350038 b(6) = c(4) = 8+13+15 = 36 = a(4).
%o A350038 (Ruby)
%o A350038 def A(n)
%o A350038   ary = []
%o A350038   (1..n).each{|i|
%o A350038     (i + 1..n).each{|j|
%o A350038       if i.gcd(j) == 1 && (i - j) % 3 > 0
%o A350038         x, y, z = j * j, i * j, i * i
%o A350038         ary << 2 * x + 5 * y + 2 * z
%o A350038         ary << 3 * x + 3 * y
%o A350038       end
%o A350038     }
%o A350038   }
%o A350038   ary
%o A350038 end
%o A350038 p A(20).uniq.sort[0..100]
%Y A350038 Cf. A024364, A201223, A264826, A350013, A350039.
%K A350038 nonn
%O A350038 1,1
%A A350038 _Seiichi Manyama_, Dec 10 2021