A306538 The least prime q such that Kronecker(D/q) = 1 where D runs through all negative fundamental discriminants (-A003657).

7, 5, 2, 3, 3, 2, 5, 3, 2, 5, 2, 3, 2, 7, 11, 2, 5, 7, 2, 3, 3, 17, 3, 2, 2, 3, 5, 2, 13, 5, 2, 2, 3, 3, 2, 7, 3, 2, 11, 11, 2, 3, 7, 5, 5, 2, 19, 2, 3, 3, 2, 41, 3, 2, 13, 3, 2, 5, 7, 2, 7, 2, 3, 5, 3, 2, 5, 2, 3, 11, 2, 31, 13, 2, 5, 2, 3, 3, 2, 5, 3, 2, 5, 23, 2, 5, 17, 7, 2, 5, 7, 2, 3, 3

Offset: 1

Author

Jianing Song, Feb 22 2019

Keywords

• nonn

Comments

a(n) is the least prime that decomposes in the imaginary quadratic field with discriminant D, D = -A003657(n).

For most n, a(n) is relatively small. There are only 472 n's among [1, 3043] (there are 3043 terms in A003657 below 10000) that violate a(n) < log(A003657(n)).

Also a(n) is the smallest prime p such that the imaginary quadratic field with discriminant D = -A003657(n) can be embedded into the p-adic field Q_p. - Jianing Song, Feb 14 2021

Examples

			Let K = Q[sqrt(-177)] with D = -708 = -A003657(218), we have: 2 and 3 divides 708, (-708/5) = (-708/7) = ... = (-708/29) = -1 and (-708/31) = +1, so 2 and 3 ramify in K, 5, 7, ..., 29 remain inert in K and 31 decomposes in K, so a(218) = 31.
		

Crossrefs

Cf. A003657.

Similar sequences: A232931, A232932 (the least prime that remains inert); A306537, this sequence (the least prime that decomposes); A306541, 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), ", ")))