A106859 - OEIS

%I #52 Jun 29 2021 18:18:56

%S 2,3,5,17,23,47,53,83,107,113,137,167,173,197,227,233,257,263,293,317,

%T 347,353,383,443,467,503,557,563,587,593,617,647,653,677,683,743,773,

%U 797,827,857,863,887,947,953,977,983,1013,1097,1103,1163,1187,1193

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

%C Discriminant=-15.

%C 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

%C 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

%C 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.]

%C From _Wolfdieter Lang_, Jun 08 2021: (Start)

%C 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.

%C 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)

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

%H Vincenzo Librandi, N. J. A. Sloane and Ray Chandler, <a href="/A106859/b106859.txt">Table of n, a(n) for n = 1..10000</a> [First 2000 terms from Vincenzo Librandi, next 691 terms from N. J. A. Sloane]

%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)

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

%o (PARI)

%o { fc(a,b,c,M) = my(p,t1,t2,n); t1 = listcreate();

%o for(n=1,M, p = prime(n);

%o t2 = qfbsolve(Qfb(a,b,c),p); if(t2 == 0,, listput(t1,p)));

%o print(t1);

%o }

%o fc(2,1,2,1000); \\ _N. J. A. Sloane_, Jun 06 2014

%Y Cf. A000003, A139827, A039949, A132235, A033212, A343241.

%K nonn,easy

%O 1,1

%A _T. D. Noe_, May 09 2005

%E Removed defective Mma program and extended the b-file using the PARI program fc. - _N. J. A. Sloane_, Jun 06 2014