A001986 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.

19, 43, 43, 67, 67, 163, 163, 163, 163, 163, 163, 222643, 1333963, 1333963, 2404147, 2404147, 20950603, 51599563, 51599563, 96295483, 96295483, 146161723, 1408126003, 3341091163, 3341091163, 3341091163, 52947440683, 52947440683, 52947440683, 193310265163

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Numbers so far are all congruent to 19 mod 24. - Ralf Stephan, Jul 07 2003

All terms are congruent to 19 mod 24. - Jianing Song, Feb 17 2019

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

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Jinyuan Wang, Table of n, a(n) for n = 1..56
• 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.
• D. H. Lehmer, E. Lehmer and D. Shanks, Integer sequences having prescribed quadratic character, Math. Comp., 24 (1970), 433-451 [Annotated scanned copy]

Crossrefs

Cf. A001987, A094845, A094846.

Cf. A001992 (the congruent to 5 mod 8 case), A094851, A094852, A094853.

See A094841, A094842, A094843, A094844 for the case where the terms are not restricted to the primes.

Programs

PARI

isok(p, oddpn) = {forprime(q=3, oddpn, if (kronecker(-p, q) != -1, return (0));); return (1);}
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

Extensions

Revised by N. J. A. Sloane, Jun 14 2004

a(28)-a(30) from Jinyuan Wang, Apr 09 2020