A306541 - OEIS

%I #13 Mar 31 2019 02:20:27

%S 2,5,2,2,3,2,3,2,2,5,2,3,2,2,2,7,2,2,2,3,2,3,2,2,7,3,2,2,2,3,2,5,2,2,

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

%U 5,3,2,2,2,3,2,2,2,3,2,2,2,2,5,2,3,2,2,2

%N The least prime q such that Kronecker(D/q) >= 0 where D runs through all positive fundamental discriminants (A003658).

%C a(n) is the least prime that either decomposes or ramifies in the real quadratic field with discriminant D, D = A003658(n).

%C For most n, a(n) is relatively small. There are only 86 n's among [1, 3044] (there are 3044 terms in A003658 below 10000) that violate a(n) < log(A003658(n)).

%e Let K = Q[sqrt(293)] with D = 293 = A003658(90), we have: (293/2) = (293/3) = ... = (293/13) = -1 and (293/17) = +1, so 2, 3, 5, 7, 11 and 13 remain inert in K and 17 decomposes in K, so a(90) = 17.

%o (PARI) b(D)=forprime(p=2, oo, if(kronecker(D, p)>=0, return(p)))

%o for(n=1, 300, if(isfundamental(n), print1(b(n), ", ")))

%Y Cf. A003658.

%Y Similar sequences: A232931, A232932 (the least prime that remains inert); A306537, A306538 (the least prime that decomposes); this sequence, A306542 (the least prime that decomposes or ramifies).

%K nonn

%O 1,1

%A _Jianing Song_, Feb 22 2019