

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.


21



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
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OFFSET

1,1


COMMENTS

If p>2 is a member then it has a unique representation as x^2+2y^2 [Frei, Theorem 3].  N. J. A. Sloane, May 30 2014
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). [From 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.
G. Frei, Euler's convenient numbers, Math. Intell. Vol. 7, No. 3 (1985), 5558 and 64.


LINKS

T. D. Noe, Table of n, a(n) for n=1..1000
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

QuadPrimes[1, 0, 2, 10000] (* see A106856 *) Note: the original QuadPrimes had a bug which could sometimes give wrong answers. This sequence should be checked (unless the coefficient of xy in the quadratic form is zero, in which case QuadPrimes gives correct answers).  N. J. A. Sloane, Jun 04 2014
Select[Prime[Range[200]], MemberQ[{1, 2, 3}, Mod[#, 8]]&] (* Harvey P. Dale, Mar 16 2013 *)


PROG

(Haskell)
a033203 n = a033203_list !! (n1)
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


CROSSREFS

Cf. A033200.
Cf. A039706, A003628 (complement with respect to A000040).
Cf. A002332, A002333.  Zak Seidov, May 29 2014
Primes in A002479.
Sequence in context: A091734 A038902 A019355 * A051100 A051088 A051092
Adjacent sequences: A033200 A033201 A033202 * A033204 A033205 A033206


KEYWORD

nonn,nice,easy


AUTHOR

N. J. A. Sloane.


STATUS

approved



