A001988 Let p be the n-th odd prime. a(n) is the least prime congruent to 7 modulo 8 such that Legendre(-a(n), q) = -Legendre(-1, q) for all odd primes q <= p.

7, 7, 127, 463, 463, 487, 1423, 33247, 73327, 118903, 118903, 118903, 454183, 773767, 773767, 773767, 773767, 86976583, 125325127, 132690343, 788667223, 788667223, 1280222287, 2430076903, 10703135983, 10703135983, 10703135983

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Numbers so far are all congruent to 7 (mod 24). - Ralf Stephan, Jul 07 2003

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

• 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. A001990.

Programs

PARI

isok(p, oddpn) = {forprime(q=3, oddpn, if (kronecker(p, q) != -kronecker(-1, q), return (0));); return (1);}
a(n) = {oddpn = prime(n+1); forprime(p=3, , if ((p%8) == 7, if (isok(p, oddpn), return (p));););} \\ Michel Marcus, Oct 18 2017

Python

from sympy import legendre_symbol as L, primerange, prime, nextprime
def isok(p, oddpn):
    for q in primerange(3, oddpn + 1):
        if L(p, q)!=-L(-1, q): return 0
    return 1
def a(n):
    oddpn=prime(n + 1)
    p=3
    while True:
        if p%8==7:
            if isok(p, oddpn): return p
        p=nextprime(p) # Indranil Ghosh, Oct 23 2017, after PARI code by Michel Marcus

Extensions

Better name and more terms from Sean A. Irvine, Mar 06 2013

Name and offset corrected by Michel Marcus, Oct 18 2017