cp's OEIS Frontend

A220171 An ordered subset of primitive values of x^2 + x*y + y^2 where at least two ordered pairs (x1,y1) and (x2,y2) with x1 != x2, y1 != y2 and gcd(x1,y1) = gcd(x2,y2) = 1 yield identical primitive values.

%I A220171 #22 Aug 02 2019 03:42:42
%S A220171 91,133,217,247,259,273,301,399,403,427,469,481,511,553,559,589,637,
%T A220171 651,679,703,721,741,763,777,793,817,871,889,903,931,949,973,1027,
%U A220171 1057,1099,1141,1147,1159,1183,1209,1261,1267,1273,1281,1333,1339,1351
%N A220171 An ordered subset of primitive values of x^2 + x*y + y^2 where at least two ordered pairs (x1,y1) and (x2,y2) with x1 != x2, y1 != y2 and gcd(x1,y1) = gcd(x2,y2) = 1 yield identical primitive values.
%C A220171 The primitive values of x^2 + x*y + y^2 where x >= y >= 0 and gcd(x, y) = 1 are given by A034017. However there are incidents in the sequence A034017 where different values of (x, y) yield the same primitive value. Furthermore, the number of solutions for a given primitive value equates to a power of 2. See A121940.
%F A220171 n such that n = x1^2 + x1*y1 + y1^2 = x2^2 + x2*y2 + y2^2 with x1 != x2, y1 != y2 and gcd(x1,y1) = gcd(x2,y2) = 1.
%e A220171 a(3) = 217 because it is the 3rd incident in ascending order of the primitive x^2 + x*y + y^2 that yields multiple solutions. This happens when (x, y) = (9, 8) and (13, 3).
%t A220171 maxLen = 100; sol[k_] := Solve[m^2 + m*n + n^2 == k && m > n > 0 && GCD[m, n] == 1, Integers]; getlist[l_] := Which[Length[sol[l]] == 0, {}, True, {m, n} /. sol[l]]; list = {}; p = 1; While[Length[list] < maxLen, (While[Length[getlist[p]] < 2, p++]; list = Append[list, p]; p++)]; list
%Y A220171 Cf. A034017, A121940.
%K A220171 nonn
%O A220171 1,1
%A A220171 _Frank M Jackson_, Dec 06 2012