

A106859


Primes of the form 2x^2 + xy + 2y^2.


8



2, 3, 5, 17, 23, 47, 53, 83, 107, 113, 137, 167, 173, 197, 227, 233, 257, 263, 293, 317, 347, 353, 383, 443, 467, 503, 557, 563, 587, 593, 617, 647, 653, 677, 683, 743, 773, 797, 827, 857, 863, 887, 947, 953, 977, 983, 1013, 1097, 1103, 1163, 1187, 1193
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OFFSET

1,1


COMMENTS

Discriminant=15.
If p is a prime >= 17 in this sequence then k==0 (mod 4) for all k satisfying "B(2k)(p^k1) is an integer" where B are the Bernoulli numbers.  Benoit Cloitre, Nov 14 2005
Equals {2, 3, 5 and primes congruent to 17, 23 (mod 30)}; see A039949 and A132235. Except for 2, the same as primes of the form 3x^2 + 5y^2, which has discriminant 60.  T. D. Noe, May 02 2008
Equals {3, 5 and primes congruent to 2, 8 (mod 15)} sorted; see A033212. This form is in the only nonprincipal class (respectively, genus) for fundamental discriminant 15.  Rick L. Shepherd, Jul 25 2014 [See A343241 for the 2, 8 (mod 15) primes.]
From Wolfdieter Lang, Jun 08 2021: (Start)
Regarding the above comment of T. D. Noe on the form [3, 0, 5]: the class number h(60) = 2 = A000003(15), and [1, 0, 15] is the principal reduced form, representing the primes given in A033212.
The form [3, 0, 5] represents the proper equivalence class of the second genus of forms of discriminant Disc = 60. The Legendre symbol for the odd primes, not 3 or 5, satisfy L(3p) = 1 and L(5p) = 1, leading to primes p == {17, 23, 47, 53} (mod 60). See the Buell reference, p. 52, for the two characters L(p3) and L(p5). The prime 2 is represented by the imprimitive reduced form [2, 2, 8] of Disc = 60. (End)


REFERENCES

D. A. Buell, Binary Quadratic Forms. SpringerVerlag, NY, 1989, pp. 5152.


LINKS

Vincenzo Librandi, N. J. A. Sloane and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 2000 terms from Vincenzo Librandi, next 691 terms from N. J. A. Sloane]
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)


MATHEMATICA

QuadPrimes2[2, 1, 2, 100000] (* see A106856 *)


PROG

(PARI)
{ fc(a, b, c, M) = my(p, t1, t2, n); t1 = listcreate();
for(n=1, M, p = prime(n);
t2 = qfbsolve(Qfb(a, b, c), p); if(t2 == 0, , listput(t1, p)));
print(t1);
}
fc(2, 1, 2, 1000); \\ N. J. A. Sloane, Jun 06 2014


CROSSREFS

Cf. A000003, A139827, A039949, A132235, A033212, A343241.
Sequence in context: A180474 A155978 A235925 * A055472 A077499 A127061
Adjacent sequences: A106856 A106857 A106858 * A106860 A106861 A106862


KEYWORD

nonn,easy


AUTHOR

T. D. Noe, May 09 2005


EXTENSIONS

Removed defective Mma program and extended the bfile using the PARI program fc.  N. J. A. Sloane, Jun 06 2014


STATUS

approved



