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The least positive integer k such that Kronecker(D/k) = -1 where D runs through all negative fundamental discriminants (-A003657).
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%I #28 Mar 08 2020 08:30:20

%S 2,3,3,5,2,7,2,11,5,13,3,2,7,3,2,5,2,3,3,11,2,2,5,7,3,2,13,5,3,2,7,3,

%T 11,2,11,2,7,11,7,2,3,2,5,3,2,5,3,3,5,2,11,2,13,5,5,2,5,3,2,7,2,3,2,2,

%U 5,13,2,3,2,5,17,3,2,7,3,3,5,2,13,2,7,5,19,2,3,11,3,2,5,2,3,3,7,2,5,2,5,11,5,3,2,5,3,2,11,2,3,7,2,2,11,7,3,2,5,3,2,5,3,3,2,11,2,19,5,5,2,3,2,17,3,2,7,2,3,3,13,2,5,2,5,11,7,3,2,7,3,13,2,3,5,2,2

%N The least positive integer k such that Kronecker(D/k) = -1 where D runs through all negative fundamental discriminants (-A003657).

%C From _Jianing Song_, Feb 14 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 imaginary quadratic field with discriminant D, D = -A003657(n). (End)

%H Amiram Eldar, <a href="/A232932/b232932.txt">Table of n, a(n) for n = 1..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 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.

%F a(n) = A306220(A003657(n)). - _Jianing Song_, Feb 14 2019

%e A003657(4) = 8, (-8/5) = -1, (-8/3) = 1 and (-8/2) = 0, so a(4) = 5.

%t nMax = 200; FundamentalDiscriminantQ[n_] := n != 1 && (Mod[n, 4] == 1 || ! Unequal[Mod[n, 16], 8, 12]) && SquareFreeQ[n/2^IntegerExponent[n, 2]]; discrims = Select[-Range[4 nMax], FundamentalDiscriminantQ]; f[d_] := For[k = 1, True, k++, If[FreeQ[{0, 1}, KroneckerSymbol[d, k]], Return[k] ] ]; a[n_] := f[discrims[[n]]]; Table[a[n], {n, 1, nMax}] (* _Jean-François Alcover_, Nov 05 2016, after _Robert G. Wilson v_ *)

%Y Cf. A003657, A306220.

%K nonn

%O 1,1

%A _Steven Finch_, Dec 02 2013

%E Name simplified by _Jianing Song_, Feb 14 2019