A094847 Let p = n-th odd prime. Then a(n) = least positive integer congruent to 5 modulo 8 such that Legendre(a(n), q) = -1 for all odd primes q <= p.

5, 53, 173, 173, 293, 437, 9173, 9173, 24653, 74093, 74093, 74093, 170957, 214037, 214037, 214037, 2004917, 44401013, 71148173, 154554077, 154554077, 163520117, 163520117, 163520117, 261153653, 261153653, 1728061733

Offset: 1

Author

N. J. A. Sloane, Jun 14 2004

Keywords

• nonn

Comments

With an initial a(0) = 5, a(n) is the least fundamental discriminant D > 1 such that the first n + 1 primes are inert in the real quadratic field with discriminant D. See A094841 for the imaginary quadratic field case. - Jianing Song, Feb 15 2019

All terms are congruent to 5 mod 24. - Jianing Song, Feb 17 2019

Links

• Michael John Jacobson, Jr., Computational Techniques in Quadratic Fields, Master's thesis, University of Manitoba, Winnipeg, Manitoba, 1995.
• Michael John Jacobson Jr. and Hugh C. Williams, New quadratic polynomials with high densities of prime values, Math. Comp. 72 (2003), 499-519.
• D. H. Lehmer, E. Lehmer and D. Shanks, Integer sequences having prescribed quadratic character, Math. Comp., 24 (1970), 433-451.

Crossrefs

Cf. A094848, A094849, A094850.

Cf. A094841 (the imaginary quadratic field case), A094842, A094843, A094844.

See A001992, A094851, A094852, A094853 for the case where the terms are restricted to the primes.

Programs

PARI

isok(m, oddpn) = {forprime(q=3, oddpn, if (kronecker(m, q) != -1, return (0));); return (1);}
a(n) = {oddpn = prime(n+1); m = 5; while(! isok(m, oddpn), m += 8); m;} \\ Michel Marcus, Oct 17 2017