A330161 - OEIS

%I #9 Dec 08 2019 12:31:52

%S 7,8,4,3,43,88,67,148,267,760,232,1320,163,1848,45208,124195,169603,

%T 85507,121972,261627,424708,656755,35230603,80149435,154962808,

%U 289615747

%N 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.

%C 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.

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

%C 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, ...

%e 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.

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

%o a(n)=my(p=prime(n)); for(D=3, oo, if(isfundamental(-D) && b(-D)==p, return(D)))

%Y Cf. A306538, A330221.

%K nonn,more

%O 1,1

%A _Jianing Song_, Dec 03 2019