A094841 - OEIS

%I #36 Feb 21 2019 15:08:24

%S 19,43,43,67,67,163,163,163,163,163,163,77683,77683,1333963,2404147,

%T 2404147,20950603,36254563,51599563,96295483,96295483,114148483,

%U 269497867,269497867,269497867,269497867,585811843,52947440683

%N 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.

%C (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

%C 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

%H William P. Orrick, <a href="/A094841/b094841.txt">Table of n, a(n) for n = 1..58</a> (first 28 terms from N. J. A. Sloane)

%H M. J. Jacobson, Jr., <a href="http://pages.cpsc.ucalgary.ca/~jacobs/PDF/mthesis.pdf">Computational Techniques in Quadratic Fields</a>, Master's thesis, University of Manitoba, Winnipeg, Manitoba, 1995.  (This sequence is given in Table 6.6.)

%H Michael John Jacobson Jr. and Hugh C. Williams, <a href="https://doi.org/10.1090/S0025-5718-02-01418-7">New quadratic polynomials with high densities of prime values</a>, Math. Comp. 72 (2003), 499-519.

%H D. H. Lehmer, E. Lehmer and D. Shanks, <a href="https://doi.org/10.1090/S0025-5718-1970-0271006-X">Integer sequences having prescribed quadratic character</a>, Math. Comp., 24 (1970), 433-451.

%F a(n) = 4*A181667(n+1) - 1. - _William P. Orrick_, Mar 19 2017

%o (PARI) isok(m, oddpn) = {forprime(q=3, oddpn, if (kronecker(-m, q) != -1, return (0));); return (1);}

%o a(n) = {oddpn = prime(n+1); m = 3; while(! isok(m, oddpn), m += 8); m;} \\ _Michel Marcus_, Oct 17 2017

%Y Cf. A094842, A094843, A094844.

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

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

%Y Cf. also A181667.

%K nonn

%O 1,1

%A _N. J. A. Sloane_, Jun 13 2004