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A357275 Smallest side of integer-sided primitive triangles whose angles satisfy A < B < C = 2*Pi/3.

%I A357275 #23 Sep 25 2022 22:56:03
%S A357275 3,7,5,11,7,13,16,9,32,17,40,11,19,55,40,24,13,23,65,69,56,25,75,15,
%T A357275 104,32,56,29,17,87,85,119,31,72,93,64,144,19,95,133,40,136,35,105,21,
%U A357275 105,37,111,185,88,152,176,23,80,115,161,41,123,240,48,205,240,43,25,129,175,215,88
%N A357275 Smallest side of integer-sided primitive triangles whose angles satisfy A < B < C = 2*Pi/3.
%C A357275 The triples of sides (a,b,c) with a < b < c are in nondecreasing order of largest side c, and if largest sides coincide, then by increasing order of the smallest side. This sequence lists the a's.
%C A357275 For the corresponding primitive triples and miscellaneous properties and references, see A357274.
%C A357275 Solutions a of the Diophantine equation c^2 = a^2 + a*b + b^2 with gcd(a,b) = 1 and a < b.
%C A357275 Also, a is generated by integers u, v such that gcd(u,v) = 1 and 0 < v < u, with a = u^2 - v^2.
%C A357275 This sequence is not increasing. For example, a(2) = 7 for triangle with largest side = 13 while a(3) = 5 for triangle with largest side = 19.
%C A357275 Differs from A088514, the first 20 terms are the same then a(21) = 56 while A088514(21) = 25.
%C A357275 A229858 gives all the possible values of the smallest side a, in increasing order without repetition, but for all triples, not necessarily primitive.
%C A357275 All terms of A106505 are values taken by the smallest side a, in increasing order without repetition for primitive triples, but not all the lengths of this side a are present; example: 3 is not in A106505 (see comment in A229849).
%F A357275 a(n) = A357274(n, 1).
%e A357275 a(2) = a(5) = 7 because 2nd and 5th triple are respectively (7, 8, 13) and (7, 33, 37).
%p A357275 for c from 5 to 181 by 2 do
%p A357275 for a from 3 to c-2 do
%p A357275 b := (-a + sqrt(4*c^2-3*a^2))/2;
%p A357275 if b=floor(b) and gcd(a, b)=1 and a<b then print(a); end if;
%p A357275 end do;
%p A357275 end do;
%Y A357275 Cf. A357274 (triples), this sequence (smallest side), A357276 (middle side), A357277 (largest side), A357278 (perimeter).
%Y A357275 Cf. also A002324, A050931, A088514, A106505, A229849.
%K A357275 nonn
%O A357275 1,1
%A A357275 _Bernard Schott_, Sep 23 2022