A306541 The least prime q such that Kronecker(D/q) >= 0 where D runs through all positive fundamental discriminants (A003658).

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, 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, 5, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 5, 2, 3, 2, 2, 2

Offset: 1

Author

Jianing Song, Feb 22 2019

Keywords

• nonn

Comments

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

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)).

Examples

			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.
		

Crossrefs

Cf. A003658.

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).

Programs

PARI

b(D)=forprime(p=2, oo, if(kronecker(D, p)>=0, return(p)))
for(n=1, 300, if(isfundamental(n), print1(b(n), ", ")))