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A003625 Primes congruent to {3, 5, 6} mod 7.
(Formerly M2487)
6
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified November 16 00:11 EST 2018. Contains 317252 sequences. (Running on oeis4.)