A033203 Primes p congruent to {1, 2, 3} (mod 8); or primes p of form x^2 + 2*y^2; or primes p such that x^2 = -2 has a solution mod p.

2, 3, 11, 17, 19, 41, 43, 59, 67, 73, 83, 89, 97, 107, 113, 131, 137, 139, 163, 179, 193, 211, 227, 233, 241, 251, 257, 281, 283, 307, 313, 331, 337, 347, 353, 379, 401, 409, 419, 433, 443, 449, 457, 467, 491, 499, 521, 523, 547, 563, 569, 571, 577, 587, 593, 601, 617, 619, 641, 643, 659, 673, 683

Offset

1,1

Comments

Sequence naturally partitions into two sequences: all primes p with ord_p(-2) odd (A163183, the primes dividing 2^j +1 for some odd j) and certain primes p with ord_p(-2) even (A163185). - Christopher J. Smyth, Jul 23 2009

Terms m in A047476 with A010051(m) = 1. - Reinhard Zumkeller, Dec 29 2012

References

David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989.

Links

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Formula

a(n) = A002332(n) + 2*A002333(n)^2. - Zak Seidov, May 29 2014

Mathematica

QuadPrimes2[1, 0, 2, 10000] (* see A106856 *)
Select[Prime[Range[200]], MemberQ[{1, 2, 3}, Mod[#, 8]]&] (* Harvey P. Dale, Mar 16 2013 *)

Prog

(Haskell)
a033203 n = a033203_list !! (n-1)
a033203_list = filter ((== 1) . a010051) a047476_list
-- Reinhard Zumkeller, Dec 29 2012, Jan 22 2012
(Magma) [p: p in PrimesUpTo(600) | p mod 8 in [1..3]]; // Vincenzo Librandi, Aug 11 2012
(Magma) [p: p in PrimesUpTo(800) | NormEquation(2, p) eq true]; // Bruno Berselli, Jul 03 2016
(PARI) is(n)=isprime(n) && issquare(Mod(-2, n)) \\ Charles R Greathouse IV, Nov 29 2016

Crossrefs

Cf. A033200, A002332, A002333.

Cf. A039706, A003628 (complement with respect to A000040).

Primes in A002479.

Cf. A051100 (see Mathar's comment).

Sequence in context: A341784 A038902 A019355 * A051100 A051088 A051092

Adjacent sequences: A033200 A033201 A033202 * A033204 A033205 A033206

Keyword

nonn,nice,easy

Author

N. J. A. Sloane

Status

approved