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%I M5073 N2195 #45 Apr 10 2020 11:36:09
%S 19,43,43,67,67,163,163,163,163,163,163,222643,1333963,1333963,
%T 2404147,2404147,20950603,51599563,51599563,96295483,96295483,
%U 146161723,1408126003,3341091163,3341091163,3341091163,52947440683,52947440683,52947440683,193310265163
%N Let p be the n-th odd prime. Then a(n) is the least prime congruent to 3 modulo 8 such that Legendre(-a(n), q) = -1 for all odd primes q <= p.
%C Numbers so far are all congruent to 19 mod 24. - _Ralf Stephan_, Jul 07 2003
%C All terms are congruent to 19 mod 24. - _Jianing Song_, Feb 17 2019
%C Also a(n) is the least prime r congruent to 3 mod 8 such that the first n odd primes are quadratic nonresidues modulo r. Note that r == 3 (mod 8) implies 2 is a quadratic nonresidue modulo r. See A001992 for the case where r == 5 (mod 8). - _Jianing Song_, Feb 19 2019
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Jinyuan Wang, <a href="/A001986/b001986.txt">Table of n, a(n) for n = 1..56</a>
%H Michael John Jacobson, Jr., <a href="http://hdl.handle.net/1993/18862">Computational Techniques in Quadratic Fields</a>, Master's thesis, University of Manitoba, Winnipeg, Manitoba, 1995.
%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.
%H D. H. Lehmer, E. Lehmer and D. Shanks, <a href="/A002189/a002189.pdf">Integer sequences having prescribed quadratic character</a>, Math. Comp., 24 (1970), 433-451 [Annotated scanned copy]
%o (PARI) isok(p, oddpn) = {forprime(q=3, oddpn, if (kronecker(-p, q) != -1, return (0));); return (1);}
%o a(n) = {my(oddpn = prime(n+1)); forprime(p=3, , if ((p%8) == 3, if (isok(p, oddpn), return (p));););} \\ _Michel Marcus_, Oct 17 2017
%Y Cf. A001987, A094845, A094846.
%Y Cf. A001992 (the congruent to 5 mod 8 case), A094851, A094852, A094853.
%Y See A094841, A094842, A094843, A094844 for the case where the terms are not restricted to the primes.
%K nonn
%O 1,1
%A _N. J. A. Sloane_
%E Revised by _N. J. A. Sloane_, Jun 14 2004
%E a(28)-a(30) from _Jinyuan Wang_, Apr 09 2020
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