A228251 Fundamental discriminant of least absolute value with class group of 2-rank n.

-3, 12, 60, -420, 4620, 60060, 1021020, -19399380, 446185740, 12939386460, -401120980260, -14841476269620, -608500527054420, 26165522663340060, -1229779565176982820, -65178316954380089460, 3845520700308425278140, 234576762718813941966540, -15716643102160534111758180

Offset: 0

Author

Rick L. Shepherd, Aug 18 2013

Keywords

• sign
• easy

Comments

Equivalently, fundamental discriminant of least absolute value with genus group of order 2^n (each such genus group is isomorphic to Z_2 x ... x Z_2 with exactly n copies of Z_2).

The n-th term is the product of n + 1 prime discriminants that are pairwise relatively prime. As the prime discriminants are exactly -8, -4, 8, and +-p for each odd prime p depending upon whether p == 1 (mod 4) or p == 3 (mod 4), respectively, |a(n)| = 2*A002110(n+1) for all n > 1 because usage of -4 precludes usage of +-8 (since the least such product in absolute value is wanted).

Examples

			The term a(0) = -3 because -3 is the fundamental discriminant of least absolute value whose corresponding class group, the trivial group, has 2-rank 0 (and its genus group is thus also the trivial group). Being negative, -3 is the discriminant of an imaginary quadratic field.
The term a(2) = 60 (=(-3)(-4)(5)) because its corresponding class group has 2-rank 2 (one fewer than the number of 60's prime discriminant factors); in this case the genus group is isomorphic to Z_2 x Z_2 (as the class group also happens to be here).  As 60 is positive, it is the discriminant of a real quadratic field.
		

Links

• Rick L. Shepherd, Table of n, a(n) for n = 0..348
• Index entries for sequences related to quadratic fields

Crossrefs

Cf. A003657, A003658, A002110, A100672.

Programs

PARI

{fd = -3; for(n = 0, 348, if(n > 1, pd = prime(n + 1); if(pd%4 == 3, pd = -pd); fd *= pd, if(n, fd = 12)); write("b228251.txt", n, " ", fd))}

Formula

a(n) = ((-1)^k)*2*A002110(n+1) for n > 1, where k is the number of 1 terms from A100672(3) through A100672(n+1) inclusive; a(0) = -3; a(1) = 12.