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%I #48 May 18 2022 13:29:56
%S 2,3,5,2,3,2,7,5,2,5,2,7,3,3,2,3,5,13,2,3,2,5,7,2,2,5,3,3,2,5,2,3,11,
%T 2,3,11,7,7,2,7,3,3,2,7,2,3,11,2,3,2,5,5,2,5,2,11,3,3,5,2,7,11,2,3,2,
%U 5,7,2,2,5,3,3,2,7,3,11,2,3,7,7,5,2,5,2,13,3,3,2,2,3,2,3,2,5,5,11,2,7,5,3,3,5,2,3,13,5,2,3,2,17,2,2,7,3,3,2,13,2,5,2,3,5,7,5,2,5,2,11,3,2,5,2,3,7,2,3,2,17,5,7,2,7,2,5,3,3,7,2,3,7,5,2,3
%N The least positive integer k such that Kronecker(D/k) = -1 where D runs through all positive fundamental discriminants (A003658).
%C From _Jianing Song_, Jan 30 2019: (Start)
%C a(n) is necessarily prime. Otherwise, if a(n) is not prime, we have (D/p) = 0 or 1 for all prime divisors p of a(n), so (D/a(n)) must be 0 or 1 too, a contradiction.
%C a(n) is the least inert prime in the real quadratic field with discriminant D, D = A003658(n). (End)
%H Charles R Greathouse IV, <a href="/A232931/b232931.txt">Table of n, a(n) for n = 2..10000</a>
%H S. R. Finch, <a href="/A232927/a232927.pdf">Average least nonresidues</a>, December 4, 2013. [Cached copy, with permission of the author]
%H Andrew Granville, R. A. Mollin and H. C. Williams, <a href="https://doi.org/10.4153/CJM-2000-017-5">An upper bound on the least inert prime in a real quadratic field</a>, Canad. J. Math. 52:2 (2000), pp. 369-380.
%H P. Pollack, <a href="http://dx.doi.org/10.1016/j.jnt.2011.12.015">The average least quadratic nonresidue modulo m and other variations on a theme of Erdős</a>, J. Number Theory 132 (2012) 1185-1202.
%H Enrique Treviño, <a href="http://campus.lakeforest.edu/trevino/LeastInertPrimeMCOM.pdf">The least inert prime in a real quadratic field</a>, Mathematics of Computation 81:279 (2012), pp. 1777-1797. See also his <a href="http://campus.lakeforest.edu/trevino/LeastInertPrimeTalkUSC.pdf">PANTS 2010 talk</a>.
%F With D = A003658(n): Mollin conjectured, and Granville, Mollin, & Williams proved, that for n > 1128, a(n) <= D^0.5 / 2. Treviño proves that for n > 484, a(n) <= D^0.45. Asymptotically the best known upper bound for the exponent is less than 0.16 when D is prime and 1/4 + epsilon (for any epsilon > 0) for general D. - _Charles R Greathouse IV_, Apr 23 2014 (corrected by _Enrique Treviño_, Mar 18 2022)
%F a(n) = A092419(A003658(n) - floor(sqrt(A003658(n))), n >= 2. - _Jianing Song_, Jan 30 2019
%e A003658(3) = 8, (8/3) = -1 and (8/2) = 0, so a(3) = 3.
%t nMax = 200; A003658 = Select[Range[4nMax], NumberFieldDiscriminant[Sqrt[#]] == #&]; f[d_] := For[k = 1, True, k++, If[FreeQ[{0, 1}, KroneckerSymbol[d, k]], Return[k]]]; a[n_] := f[A003658[[n]]]; Table[a[n], {n, 2, nMax}] (* _Jean-François Alcover_, Nov 05 2016 *)
%o (PARI) lp(D)=forprime(p=2,,if(kronecker(D,p)<0,return(p)))
%o for(n=5,1e3,if(isfundamental(n),print1(lp(n)", "))) \\ _Charles R Greathouse IV_, Apr 23 2014
%Y Cf. A003658, A092419, A241482.
%K nonn
%O 2,1
%A _Steven Finch_, Dec 02 2013
%E Name simplified by _Jianing Song_, Jan 30 2019
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