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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 (list; graph; refs; listen; history; text; internal format)
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^k-1) 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 non-principal 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(-3|p) = -1 and L(5|p) = -1, leading to primes p == {17, 23, 47, 53} (mod 60). See the Buell reference, p. 52, for the two characters L(p|3) and L(p|5). The prime 2 is represented by the imprimitive reduced form [2, 2, 8] of Disc = -60. (End)

REFERENCES

D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, pp. 51-52.

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 b-file using the PARI program fc. - N. J. A. Sloane, Jun 06 2014

STATUS

approved

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Last modified January 22 05:19 EST 2022. Contains 350481 sequences. (Running on oeis4.)