Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%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