A003625 Primes congruent to {3, 5, 6} mod 7. (Formerly M2487)

3, 5, 13, 17, 19, 31, 41, 47, 59, 61, 73, 83, 89, 97, 101, 103, 131, 139, 157, 167, 173, 181, 199, 223, 227, 229, 241, 251, 257, 269, 271, 283, 293, 307, 311, 313, 349, 353, 367, 383, 397, 409, 419, 433, 439, 461, 467, 479, 503, 509, 521, 523, 563, 577, 587, 593

Offset

1,1

Comments

Inert rational primes in Q(sqrt(-7)).

For terms >= 13, sequence consists of primes p such that Sum_{k=0..p} binomial(2*k,k)^3) == 8 (mod p). - Benoit Cloitre, Aug 10 2003

Primes which cannot be written in the form a^2 + 7*b^2, where a >= 0, b >= 0. - V. Raman, Sep 08 2012

Conjecture: Also such primes p where the polynomial x^2 + x + 2 is irreducible over GF(p). - Federico Provvedi, Jul 21 2018

Primes that have -7 as a quadratic nonresidue, or equivalently, primes that are quadratic nonresidues modulo 7. - Jianing Song, Jul 21 2018

References

H. Hasse, Number Theory, Springer-Verlag, NY, 1980, p. 498.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Index to sequences related to decomposition of primes in quadratic fields

Mathematica

Select[Prime[Range[800]], MemberQ[{3, 5, 6}, Mod[#, 7]]&] (* Vincenzo Librandi, Aug 04 2012 *)

Prog

(Magma) [p: p in PrimesUpTo(1000) | p mod 7 in [3, 5, 6]]; // Vincenzo Librandi, Aug 04 2012
(PARI)  {a(n) = local( cnt, m ); if( n<1, return( 0 )); while( cnt < n, if( isprime( m++) && kronecker( -7, m )==-1, cnt++ )); m} /* Michael Somos, Aug 14 2012 */

Crossrefs

Sequence in context: A184796 A180944 A049282 * A105900 A260191 A094745

Adjacent sequences: A003622 A003623 A003624 * A003626 A003627 A003628

Keyword

nonn

Author

N. J. A. Sloane, Mira Bernstein

Status

approved