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A232932 The least positive integer k such that Kronecker(D/k) = -1 where D runs through all negative fundamental discriminants (-A003657).

%I A232932 #28 Mar 08 2020 08:30:20
%S A232932 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 A232932 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 A232932 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 A232932 The least positive integer k such that Kronecker(D/k) = -1 where D runs through all negative fundamental discriminants (-A003657).
%C A232932 From _Jianing Song_, Feb 14 2019: (Start)
%C A232932 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 A232932 a(n) is the least inert prime in the imaginary quadratic field with discriminant D, D = -A003657(n). (End)
%H A232932 Amiram Eldar, <a href="/A232932/b232932.txt">Table of n, a(n) for n = 1..10000</a>
%H A232932 S. R. Finch, <a href="/A232927/a232927.pdf">Average least nonresidues</a>, December 4, 2013. [Cached copy, with permission of the author]
%H A232932 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 A232932 a(n) = A306220(A003657(n)). - _Jianing Song_, Feb 14 2019
%e A232932 A003657(4) = 8, (-8/5) = -1, (-8/3) = 1 and (-8/2) = 0, so a(4) = 5.
%t A232932 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 A232932 Cf. A003657, A306220.
%K A232932 nonn
%O A232932 1,1
%A A232932 _Steven Finch_, Dec 02 2013
%E A232932 Name simplified by _Jianing Song_, Feb 14 2019