A094841 Let p = n-th odd prime. Then a(n) = least positive integer congruent to 3 modulo 8 such that Legendre(-a(n), q) = -1 for all odd primes q <= p.

19, 43, 43, 67, 67, 163, 163, 163, 163, 163, 163, 77683, 77683, 1333963, 2404147, 2404147, 20950603, 36254563, 51599563, 96295483, 96295483, 114148483, 269497867, 269497867, 269497867, 269497867, 585811843, 52947440683

Offset: 1

Author

N. J. A. Sloane, Jun 13 2004

Keywords

• nonn

Comments

(a(n-1) + 1)/4 is the least positive integer c such that x^2 + x + c is not divisible by the first n primes. This implies that a(n) is congruent to 19 mod 24 and that a(n) is congruent to 43 or 67 mod 120 for n > 1. - William P. Orrick, Mar 19 2017

With an initial a(0) = 3, a(n) is the negated fundamental discriminant D < 0 with the least absolute value such that the first n + 1 primes are inert in the imaginary quadratic field with discriminant D. See A094847 for the real discriminant case. - Jianing Song, Feb 15 2019

Links

• William P. Orrick, Table of n, a(n) for n = 1..58 (first 28 terms from N. J. A. Sloane)
• M. J. Jacobson, Jr., Computational Techniques in Quadratic Fields, Master's thesis, University of Manitoba, Winnipeg, Manitoba, 1995. (This sequence is given in Table 6.6.)
• 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. A094842, A094843, A094844.

Cf. A094847 (the real quadratic field case), A094848, A094849, A094850.

See A001986, A001987, A094845, A094846 for the case where the terms are restricted to the primes.

Cf. also A181667.

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 = 3; while(! isok(m, oddpn), m += 8); m;} \\ Michel Marcus, Oct 17 2017

Formula

a(n) = 4*A181667(n+1) - 1. - William P. Orrick, Mar 19 2017