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.

91, 133, 217, 247, 259, 273, 301, 399, 403, 427, 469, 481, 511, 553, 559, 589, 637, 651, 679, 703, 721, 741, 763, 777, 793, 817, 871, 889, 903, 931, 949, 973, 1027, 1057, 1099, 1141, 1147, 1159, 1183, 1209, 1261, 1267, 1273, 1281, 1333, 1339, 1351

Offset: 1

Author

Frank M Jackson, Dec 06 2012

Keywords

• nonn

Comments

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.

Examples

			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).
		

Crossrefs

Cf. A034017, A121940.

Programs

Mathematica

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

Formula

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.