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Smallest prime p > prime(n+2) such that p is a quadratic residue mod the first n odd primes 3, 5, 7, 11, ..., prime(n+1), and p is a quadratic non-residue mod prime(n+2).
9

%I #36 Mar 07 2013 20:33:16

%S 5,7,19,79,331,751,1171,7459,10651,18379,90931,78439,399499,644869,

%T 2631511,1427911,4355311,5715319,49196359,43030381,163384621,

%U 249623581,452980999,1272463669,505313251

%N Smallest prime p > prime(n+2) such that p is a quadratic residue mod the first n odd primes 3, 5, 7, 11, ..., prime(n+1), and p is a quadratic non-residue mod prime(n+2).

%C Same as smallest prime p with property that the Legendre symbol (p|q) = 1 for the first n odd primes q = prime(k+1), k = 1, 2, ..., n, and (p|q) = -1 for q = prime(n+2). - _T. D. Noe_, Mar 06 2013

%e Let f(p) = list of Legendre(p|q) for q = 3,5,7,11,13,...

%e Then f(3), f(5), f(7), f(11), ... are:

%e p=3: 0, -1, -1, 1, 1, -1, -1, 1, -1, -1, 1, -1, -1, 1, -1, 1, 1, -1, 1, 1, -1, 1, -1, ...

%e p=5: -1, 0, -1, 1, -1, -1, 1, -1, 1, 1, -1, 1, -1, -1, -1, 1, 1, -1, 1, -1, 1, -1, 1, ...

%e p=7: 1, -1, 0, -1, -1, -1, 1, -1, 1, 1, 1, -1, -1, 1, 1, 1, -1, -1, -1, -1, -1, 1, -1, ...

%e p=11: -1, 1, 1, 0, -1, -1, 1, -1, -1, -1, 1, -1, 1, -1, 1, -1, -1, -1, -1, -1, 1, 1, 1, ...

%e p=13: 1, -1, -1, -1, 0, 1, -1, 1, 1, -1, -1, -1, 1, -1, 1, -1, 1, -1, -1, -1, 1, -1, -1, ...

%e p=17: -1, -1, -1, -1, 1, 0, 1, -1, -1, -1, -1, -1, 1, 1, 1, 1, -1, 1, -1, -1, -1, 1, 1, ...

%e p=19: 1, 1, -1, -1, -1, 1, 0, -1, -1, 1, -1, -1, -1, -1, -1, 1, 1, 1, 1, 1, 1, -1, -1, ...

%e p=5 is the first list that begins with -1, so a(0) = 5,

%e p=7 is the first list that begins 1, -1, so a(1) = 7,

%e p=19 is the first list that begins 1, 1, -1, so a(2) = 19.

%t f[n_] := Block[{k = 2}, While[ JacobiSymbol[n, Prime[k]] == 1, k++ ]; Prime[k]]; t = Table[0, {50}]; Do[p = Prime[n]; a = f[p]; If[ t[[ PrimePi[a]]] == 0, t[[ PrimePi[a]]] = p; Print[ PrimePi[a], " = ", p]], {n, 10^9}]

%Y Cf. A094929, A222756 (p and q switched).

%Y See also A096637, A096638, A096639, A096640. - _Jonathan Sondow_, Mar 07 2013

%K nonn

%O 0,1

%A _Robert G. Wilson v_, Jun 24 2004

%E Better definition from _T. D. Noe_, Mar 06 2013

%E Entry revised by _N. J. A. Sloane_, Mar 06 2013

%E Simpler definition from _Jonathan Sondow_, Mar 06 2013