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Perimeters of more than one primitive 120-degree integer triangle.
3

%I #15 Dec 12 2021 02:16:14

%S 2730,4080,7590,9044,11704,12180,13020,13485,13920,14880,15810,16100,

%T 18870,21090,22755,23370,24752,25172,26445,27060,28380,29670,30315,

%U 31020,32430,32890,33810,34545,34580,36660,37950,38038,38220,38955,41340,42476,44520,46046,46110

%N Perimeters of more than one primitive 120-degree integer triangle.

%H Seiichi Manyama, <a href="/A350047/b350047.txt">Table of n, a(n) for n = 1..10000</a>

%e 897^2 + 560^2 + 897*560 = 1273^2, 168^2 + 1235^2 + 168*1235 = 1327^2 and 897 + 560 + 1273 = 168 + 1235 + 1327 = 2730. So 2730 is a term.

%e 38640^2 + 5291^2 + 38640*5291 = 41539^2, 23088^2 + 22715^2 + 23088*22715 = 39667^2, 10857^2 + 34040^2 + 10857*34040 = 40573^2 and 38640 + 5291 + 41539 = 23088 + 22715 + 39667 = 10857 + 34040 + 40573 = 85470. So 85470 is a term.

%o (Ruby)

%o def A(n)

%o ary = []

%o (1..n).each{|i|

%o (i + 1..n).each{|j|

%o if i.gcd(j) == 1 && (i - j) % 3 > 0

%o ary << 2 * j * j + 3 * i * j + i * i

%o end

%o }

%o }

%o ary

%o end

%o p A(200).group_by(&:to_i).select{|k, v| v.size > 1}.keys.sort[0..50]

%Y Cf. A350039, A350045.

%K nonn

%O 1,1

%A _Seiichi Manyama_, Dec 11 2021