A121243 Primes of the form 4*x^2 + 4*x*y + 9*y^2.

17, 73, 89, 97, 193, 233, 241, 281, 401, 433, 449, 601, 617, 641, 673, 769, 929, 937, 977, 1009, 1033, 1049, 1097, 1193, 1289, 1297, 1361, 1409, 1433, 1481, 1489, 1609, 1697, 1721, 1753, 1801, 1873, 1913

Offset: 1

Author

Steven Finch, Aug 22 2006

Keywords

• nonn

Comments

This sequence is complementary to A105389 in the sense that the two sequences are disjoint and their union constitutes all primes p satisfying Mod[p,8]=1.

Primes satisfying Mod[p,8]=1 are of form x^2+8y^2 (A007519), with the sequence above as odd y, while A105389 is even y. This can be seen by expressing the former as (2x+y)^2+8y^2 (where y can only be odd), while the latter is u^2+8(2v)^2. [From Tito Piezas III, Jan 01 2009]

Examples

			17 = 4*1^2 + 4*1*1 + 9*1^2, 73 = 4*1^2 + 4*1*(-3) + 9*(-3)^2
		

Links

• Vincenzo Librandi and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi]
• S. R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.
• Claudio Qureshi, Antonio Campello, Sueli I. R. Costa, Non-Existence of Linear Perfect Lee Codes With Radius 2 for Infinitely Many Dimensions, IEEE Transactions on Information Theory (2018) Vol. 64, Issue 4, pp. 3042-3047.
• N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Crossrefs

Cf. A105389.

Programs

Mathematica

QuadPrimes2[4, -4, 9, 10000] (* see A106856 *)
(* Second program: *)
max = 2000; Table[yy = {y, Floor[-2x/9 - 1/9 Sqrt[9max - 32x^2]], Ceiling[-2x/9 + 1/9 Sqrt[9max - 32x^2]]}; Table[4x^2 + 4 x y + 9y^2, yy // Evaluate], {x, 0, Ceiling[3Sqrt[max]/(4Sqrt[2])]}] // Flatten // Union // Select[#, # <= max && PrimeQ[#]&]& // Quiet (* Jean-François Alcover, Oct 08 2018 *)