cp's OEIS Frontend

A357276 Middle side of integer-sided primitive triangles whose angles satisfy A < B < C = 2*Pi/3 = 120 degrees.

%I A357276 #11 Sep 30 2022 23:43:01
%S A357276 5,8,16,24,33,35,39,56,45,63,51,85,80,57,77,95,120,120,88,91,115,143,
%T A357276 112,161,105,175,165,195,208,160,168,145,224,203,187,221,155,261,217,
%U A357276 192,279,209,288,247,320,272,323,280,231,315,273,259,385,357,333,304,399,352,253,407,299,287,440
%N A357276 Middle side of integer-sided primitive triangles whose angles satisfy A < B < C = 2*Pi/3 = 120 degrees.
%C A357276 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 b's.
%C A357276 For the corresponding primitive triples and miscellaneous properties and references, see A357274.
%C A357276 Solutions b of the Diophantine equation c^2 = a^2 + a*b + b^2 with gcd(a,b) = 1 and a < b.
%C A357276 Also, b is generated by integers u, v such that gcd(u,v) = 1 and 0 < v < u, with b = 2*u*v + v^2.
%C A357276 This sequence is not increasing. For example, a(8) = 56 for triangle with largest side c = 61  while a(9) = 45 for triangle with largest side c = 67.
%C A357276 Differs from A088586, the first 20 terms are the same then a(21) = 115 while A088586(21) = 143.
%C A357276 A229849 gives all the possible values of the middle side b, in increasing order without repetition, for primitive triples, while A229859 gives all the possible values of the middle side b, in increasing order without repetition, but for all triples, not necessarily primitive.
%e A357276 a(17) = a(18) = 120 since 17th and 18th triples are respectively (13, 120, 127) and (23, 120, 133).
%p A357276 for c from 5 to 500 by 2 do
%p A357276 for a from 3 to c-2 do
%p A357276 b := (-a + sqrt(4*c^2-3*a^2))/2;
%p A357276 if b=floor(b) and gcd(a,b)=1 and a<b then print(b); end if;
%p A357276 end do;
%p A357276 end do;
%Y A357276 Cf. A357274 (triples), A357275 (smallest side), this sequence (middle side), A357277 (largest side), A357278 (perimeter).
%Y A357276 Cf. also A088586, A229849, A229859.
%K A357276 nonn
%O A357276 1,1
%A A357276 _Bernard Schott_, Sep 25 2022