A001992 Let p = n-th odd prime. Then a(n) = least prime congruent to 5 modulo 8 such that Legendre(a(n), q) = -1 for all odd primes q <= p. (Formerly M4012 N1663)

5, 53, 173, 173, 293, 2477, 9173, 9173, 61613, 74093, 74093, 74093, 170957, 360293, 679733, 2004917, 2004917, 69009533, 138473837, 237536213, 384479933, 883597853, 1728061733, 1728061733, 1728061733, 1728061733

Offset

1,1

Comments

All terms are congruent to 5 mod 24. - Jianing Song, Feb 17 2019

Also a(n) is the least prime r congruent to 5 mod 8 such that the first n odd primes are quadratic nonresidues modulo r. Note that r == 5 (mod 8) implies 2 is a quadratic nonresidue modulo r. See A001986 for the case where r == 3 (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

Table of n, a(n) for n=1..26.

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. [There is an error in the table given in this paper.]

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) == 5, if (isok(p, oddpn), return (p)); ); ); } \\ Michel Marcus, Oct 17 2017

Crossrefs

Cf. A094851, A094852, A094853.

Cf. A001986 (the congruent to 3 mod 8 case), A001987, A094845, A094846.

See A094847, A094848, A094849, A094850 for the case where the terms are not restricted to the primes.

Sequence in context: A075540 A006562 A094847 * A139899 A212820 A094849

Adjacent sequences: A001989 A001990 A001991 * A001993 A001994 A001995

Keyword

nonn

Author

N. J. A. Sloane

Extensions

Corrected and extended by N. J. A. Sloane, Jun 14 2004

Status

approved