A306542 The least prime q such that Kronecker(D/q) >= 0 where D runs through all negative fundamental discriminants (-A003657).

3, 2, 2, 2, 3, 2, 5, 2, 2, 2, 2, 3, 2, 2, 11, 2, 3, 2, 2, 2, 3, 17, 2, 2, 2, 3, 2, 2, 2, 5, 2, 2, 2, 3, 2, 5, 2, 2, 2, 3, 2, 3, 2, 2, 5, 2, 2, 2, 2, 3, 2, 41, 2, 2, 2, 3, 2, 2, 7, 2, 3, 2, 3, 5, 2, 2, 3, 2, 3, 2, 2, 2, 5, 2, 2, 2, 2, 3, 2, 5, 2, 2, 2, 3, 2, 2, 2, 7

Offset: 1

Author

Jianing Song, Feb 22 2019

Keywords

• nonn

Comments

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

The quadratic field with discriminant D = -A003657(n) has class number 1 if and only if a(n) >= (1/4)*A003657(n). If the quadratic field with discriminant D = -A003657(n) has class number 3 then a(n)^2 < (1/4)*A003657(n) < a(n)^3.

For most n, a(n) is relatively small. There are only 86 n's among [1, 3043] (there are 3043 terms in A003657 below 10000) that violate a(n) < log(A003657(n)). In fact, if we ignore the first term, the only terms among the first 3043 ones that seem unusually large are a(15) = 11 (with A003657(15) = 43), a(22) = 17 (with A003657(22) = 67), a(52) = 41 (with A003657(52) = 163), a(1147) = 23 (with A003657(1147) = 3763), a(2677) = 23 (with A003657(2677) = 8803) and a(2758) = 23 (with A003657(2758) = 9067).

Examples

			Let K = Q[sqrt(-3763)] with D = -3763 = -A003657(1147), we have: (-3763/2) = (-3763/3) = ... = (-3763/19) = -1 and (-3763/23) = +1, so 2, 3, 5, 7, 11, 13, 17 and 19 remain inert in K and 23 decomposes in K, so a(1147) = 23.
		

Crossrefs

Cf. A003657.

Similar sequences: A232931, A232932 (the least prime that remains inert); A306537, A306538 (the least prime that decomposes); A306541, this sequence (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), ", ")))