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. (Formerly M5073 N2195)

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

1,1

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]

Prog

(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

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.

Sequence in context: A139580 A156897 A094841 * A270123 A139811 A095101

Adjacent sequences: A001983 A001984 A001985 * A001987 A001988 A001989

Keyword

nonn

Author

N. J. A. Sloane

Extensions

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

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

Status

approved