cp's OEIS Frontend

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.

%I A033203 #65 Jul 22 2025 15:24:19
%S A033203 2,3,11,17,19,41,43,59,67,73,83,89,97,107,113,131,137,139,163,179,193,
%T A033203 211,227,233,241,251,257,281,283,307,313,331,337,347,353,379,401,409,
%U A033203 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
%N 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.
%C A033203 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
%C A033203 Terms m in A047476 with A010051(m) = 1. - _Reinhard Zumkeller_, Dec 29 2012
%D A033203 David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989.
%H A033203 Ray Chandler, <a href="/A033203/b033203.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)
%H A033203 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)
%F A033203 a(n) = A002332(n) + 2*A002333(n)^2. - _Zak Seidov_, May 29 2014
%t A033203 QuadPrimes2[1, 0, 2, 10000] (* see A106856 *)
%t A033203 Select[Prime[Range[200]],MemberQ[{1,2,3},Mod[#,8]]&] (* _Harvey P. Dale_, Mar 16 2013 *)
%o A033203 (Haskell)
%o A033203 a033203 n = a033203_list !! (n-1)
%o A033203 a033203_list = filter ((== 1) . a010051) a047476_list
%o A033203 -- _Reinhard Zumkeller_, Dec 29 2012, Jan 22 2012
%o A033203 (Magma) [p: p in PrimesUpTo(600) | p mod 8 in [1..3]]; // _Vincenzo Librandi_, Aug 11 2012
%o A033203 (Magma) [p: p in PrimesUpTo(800) | NormEquation(2,p) eq true]; // _Bruno Berselli_, Jul 03 2016
%o A033203 (PARI) is(n)=isprime(n) && issquare(Mod(-2,n)) \\ _Charles R Greathouse IV_, Nov 29 2016
%Y A033203 Cf. A002332, A002333.
%Y A033203 Cf. A039706, A003628 (complement with respect to A000040).
%Y A033203 Primes in A002479.
%Y A033203 Cf. A051100 (see Mathar's comment).
%Y A033203 Apart from leading term the same as A033200.
%K A033203 nonn,nice,easy
%O A033203 1,1
%A A033203 _N. J. A. Sloane_