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A306537 The least prime q such that Kronecker(D/q) = 1 where D runs through all positive fundamental discriminants (A003658).

%I A306537 #17 Feb 14 2021 13:05:54
%S A306537 2,11,7,11,3,2,5,5,3,5,2,3,3,2,5,7,5,2,7,3,2,5,2,3,13,3,3,2,7,7,2,5,5,
%T A306537 2,3,2,7,3,2,3,3,2,13,5,2,5,11,5,3,2,7,11,3,13,2,3,3,2,11,2,7,2,5,3,2,
%U A306537 11,2,3,5,3,3,2,5,13,2,13,2,3,2,5,2,3,5,2
%N A306537 The least prime q such that Kronecker(D/q) = 1 where D runs through all positive fundamental discriminants (A003658).
%C A306537 a(n) is the least prime that decomposes in the real quadratic field with discriminant D, D = A003658(n).
%C A306537 For most n, a(n) is relatively small. There are only 459 n's among [1, 3044] (there are 3044 terms in A003658 below 10000) that violate a(n) < log(A003658(n)).
%C A306537 Also a(n) is the smallest prime p such that the real quadratic field with discriminant D = A003658(n) can be embedded into the p-adic field Q_p. - _Jianing Song_, Feb 14 2021
%e A306537 Let K = Q[sqrt(635)] with D = 2540 = A003658(774), we have: 2 and 5 divides 2540, (2540/3) = (2540/7) = ... = (2540/37) = -1 and (2540/41) = +1, so 2 and 5 ramify in K, 3, 7, ..., 37 remain inert in K and 41 decomposes in K, so a(774) = 41.
%o A306537 (PARI) b(D)=forprime(p=2, oo, if(kronecker(D, p)>0, return(p)))
%o A306537 for(n=1, 300, if(isfundamental(n), print1(b(n), ", ")))
%Y A306537 Cf. A003658.
%Y A306537 Similar sequences: A232931, A232932 (the least prime that remains inert); this sequence, A306538 (the least prime that decomposes); A306541, A306542 (the least prime that decomposes or ramifies).
%K A306537 nonn
%O A306537 1,1
%A A306537 _Jianing Song_, Feb 22 2019