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%I #43 Jan 22 2022 23:50:53
%S 2,3,7,23,43,47,67,83,103,107,127,163,167,223,227,263,283,307,347,367,
%T 383,443,463,467,487,503,523,547,563,587,607,643,647,683,727,743,787,
%U 823,827,863,883,887,907,947,967,983,1063,1087,1103,1123,1163,1187
%N Primes of the form 2x^2 + 2xy + 3y^2.
%C Discriminant = -20.
%C Also: Primes of the form 2x^2 - 2xy + 3y^2 with x and y nonnegative. Cf. A106864.
%C Primes congruent to 2, 3, 7 modulo 20. - _Michael Somos_, Aug 13 2006
%C In Z[sqrt(-5)], these numbers are irreducible but not prime. In terms of ideals, they generate principal ideals that are not prime (or maximal). The equation x^2 + 5y^2 = a(n) has no solutions, but x^2 = -5 (mod a(n)) does. For example, 2 * 3 = (1 - sqrt(-5))(1 + sqrt(-5)) and 7 * 23 = (9 - 4*sqrt(-5))(9 + 4*sqrt(-5)). - _Alonso del Arte_, Dec 19 2015
%D David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989; see p. 33.
%H Vincenzo Librandi and Ray Chandler, <a href="/A106865/b106865.txt">Table of n, a(n) for n = 1..10000</a> [First 1000 terms from Vincenzo Librandi]
%H N. J. A. Sloane et al., <a href="https://oeis.org/wiki/Binary_Quadratic_Forms_and_OEIS">Binary Quadratic Forms and OEIS</a> (Index to related sequences, programs, references)
%F Complement(A000040, A020669).
%e x = 1, y = 1 gives 2x^2 + 2xy + 3y^2 = 2 + 2 + 3 = 7.
%e x = 1, y = -3 gives 2x^2 + 2xy + 3y^2 = 2 - 6 + 27 = 23.
%p select(isprime, [2, seq(seq(5+s+20*i,s=[-2,2]),i=0..10^3)]); # _Robert Israel_, Dec 23 2015
%t QuadPrimes2[2, -2, 3, 10000] (* see A106856 *)
%o (PARI) is(n)=isprime(n) && #qfbsolve(Qfb(2,2,3),n)>0 \\ _Charles R Greathouse IV_, Feb 09 2017
%Y For n > 1, a(n) = A122870(n-1). Cf. A216816, A106864.
%K nonn,easy
%O 1,1
%A _T. D. Noe_, May 09 2005
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