A330161 Fundamental discriminant D < 0 with the least absolute value such that the smallest prime p such that Kronecker(D,p) = 1 is p = prime(n), negated.

7, 8, 4, 3, 43, 88, 67, 148, 267, 760, 232, 1320, 163, 1848, 45208, 124195, 169603, 85507, 121972, 261627, 424708, 656755, 35230603, 80149435, 154962808, 289615747

Offset: 1

Author

Jianing Song, Dec 03 2019

Keywords

• nonn
• more

Comments

If a(n) < (Pi*prime(n)/2)^2 (this occurs for n <= 14), then the ideal class group of Q[sqrt(-d)] necessarily has exponent <= 2. (The exponent of a group G is the smallest e > 0 such that x^e = I for all x in G, where I is the group identity.) See A330221.

It seems that lim_{n->oo} n^t/a(n) = 0 for all t > 0.

The exponent of the ideal class group of Q[sqrt(-a(n))]: 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 1, 2, 26, 16, 36, 22, 38, 24, 16, 36, 104, 388, 104, 288, ...

Examples

			D = -1848 is the fundamental discriminant D < 0 with the least absolute value such that Kronecker(D,p) <= 0 for p = 2, 3, 5, 7, ..., 41 and Kronecker(D,43) = +1, so a(14) = 1848.
		

Crossrefs

Cf. A306538, A330221.

Programs

PARI

b(D)=forprime(p=2, oo, if(kronecker(D, p)>0, return(p)))
a(n)=my(p=prime(n)); for(D=3, oo, if(isfundamental(-D) && b(-D)==p, return(D)))